Biases in Mixed-Effects Model GMMs
Nicolas Kuehn, Ken Campbell, Yousef Bozorgnia
17 June, 2024, first published 14 September 2023.
1 Introduction
This page provides code for the simulations shown in ``Use of Simulation to Identify Potential Biases in Mixed-Effects Ground-Motion Models and Variance Components’’, which highlights some biases that can occur when using point estimates of random effects/residuals in mixed effects ground-motion models. For details, see the paper. This repository is archived under < https://doi.org/10.5281/zenodo.10822835>.
We use simulations from different models and/or using different data sets to illustrate potential biases. In particular, standard deviations are underestimated when they are calculated from point estimates of random effects/residuals. For the simulations, we randomly sample event terms, site terms, and within-event/within-site residuals from their respective distributions, and then perform regessions on the sampled data to see whether we recover get the parameters used in the simulations. In this document, we generally do a single simulation for different cases, which typically highlight the points we want to made. For the paper, we repeat these simulations multiple times, since due to the relative small sample size in ground-motion data sets there ca be variability from one sample to the next.
Simulation can be a pwoeful tool to gain understanding of different
models (DeBruine and
Barr 2021). There exist several R-packages for
simulation (e.g. faux, simDesign,
simulator), but
since our models are simple, we code them up directly.
In general, a ground-motion model (GMM) can be written as \[ Y_{es} = f(\vec{c}; \vec{x}) + \delta B_e + \delta S_s + \delta WS_{es} \] or in matrix form as \[ \vec{Y} = f(\vec{c}; \mathbf{x}) + \mathbf{Z} \vec{u} + \vec{\delta WS} \] where \(\mathbf{Z}\) is the design matrix of the random effects. It is importat to remember that in general the outcome of a mixed-effects regression will give point estimates of the random effects (the conditional modes),and that there is uncetainty around them. The conditional variances of the random effects are the diagonal entries of the following matrix \[ \begin{aligned} V &= \phi_{SS}^2 \mathbf{\Lambda} \left(\mathbf{\Lambda}^T \mathbf{Z}^T \mathbf{Z} \mathbf{\Lambda} + \mathbf{I} \right)^{-1} \mathbf{\Lambda} \\ \psi(\hat{\vec{u}})^2 &= \mbox{diag}(V) \end{aligned} \] where \(\mathbf{\Lambda}\) is the relative covariance factor (Bates et al. 2015). If this uncertainty is ignored, biases can occur, as we deomstrate throughout this page. In particular, the variances of the random effects are calculated as (example for \(\tau\)) \[ \hat{\tau}^2 = \frac{1}{N_E}\sum_{i = 1}^{N_E} \widehat{\delta B}_i^2 + \frac{1}{N_E}\sum_{i = 1}^{N_E} \psi(\widehat{\delta B}_i)^2 \] which is the sum of the variance of the point estimates plus the average conditional variance. Hence, just estimating the variance (or standard deviation) of the point estimates will lead to an underestimation.
1.1 Set up
Load required libraries, and define some plot options for
ggplot2.
library(ggplot2)
library(lme4)
library(cmdstanr)
library(posterior)
library(bayesplot)
library(tidyverse)
library(INLA)
library(matrixStats)
library(latex2exp)## CmdStan path set to: /Users/nico/GROUNDMOTION/SOFTWARE/cmdstan-2.33.1theme_set(theme_bw() + theme(
axis.title = element_text(size = 30),
axis.text = element_text(size = 20),
plot.title = element_text(size = 30),
legend.text = element_text(size = 20),
legend.title = element_text(size = 20),
legend.key.width = unit(1, "cm"),
legend.box.background = element_rect(colour = "black"),
panel.grid = element_line(color = "gray",linewidth = 0.75)
))
breaks <- 10^(-10:10)
minor_breaks <- rep(1:9, 21)*(10^rep(-10:10, each=9))
lw <- 1.5 # linewidth
sp <- 4 # pointsize1.2 Example
We start with an example using real data, just to get familiar with the concepts and code. In later sections, we use simulations, which make it easy to compare the results from a regression to the true values of the parameters. We use the Ialian data from the ITA18 GMM (Lanzano et al. (2018), see also Caramenti et al. (2022)), and perform a regression on peak ground acceleration (PGA), using the functional form of ITA18.
First, we read in the data and prepare a data frame for the regression. In total, there are 4784 records from 137 events and 923 stations.
data_it <- read.csv(file.path('./Git/MixedModels_Biases/','/data','italian_data_pga_id_utm_stat.csv'))
# Set linear predictors
mh = 5.5
mref = 5.324
h = 6.924
attach(data_it)
b1 = (mag-mh)*(mag<=mh)
b2 = (mag-mh)*(mag>mh)
c1 = (mag-mref)*log10(sqrt(JB_complete^2+h^2))
c2 = log10(sqrt(JB_complete^2+h^2))
c3 = sqrt(JB_complete^2+h^2)
f1 = as.numeric(fm_type_code == "SS")
f2 = as.numeric(fm_type_code == "TF")
k = log10(vs30/800)*(vs30<=1500)+log10(1500/800)*(vs30>1500)
y = log10(rotD50_pga)
detach(data_it)
n_rec <- length(b1)
eq <- data_it$EQID
stat <- data_it$STATID
n_eq <- max(eq)
n_stat <- max(stat)
n_rec <- nrow(data_it)
data_reg <- data.frame(Y = y,
M1 = b1,
M2 = b2,
MlogR = c1,
logR = c2,
R = c3,
Fss = f1,
Frv = f2,
logVS = k,
eq = eq,
stat = stat,
intercept = 1
)
print(paste0('Number of records: ',n_rec,'; number of events: ',n_eq,'; number of stations: ',n_stat))## [1] "Number of records: 4784; number of events: 137; number of stations: 923"Now we fit the model. We fit the model using lmer,
inla, and using Stan via cmdstanr. The Stan
code can be found at https://github.com/nikuehn/MixedModels_Biases/tree/main/stan.
##################
# fit using lmer
fit_lmer <- lmer(Y ~ M1 + M2 + MlogR + logR + R + Fss + Frv + logVS + (1|eq) + (1|stat), data_reg)
tmp <- as.data.frame(VarCorr(fit_lmer))$sdcor
phi_s2s_lmer <- tmp[1]
tau_lmer <- tmp[2]
phi_ss_lmer <- tmp[3]
deltaB <- ranef(fit_lmer)$eq$`(Intercept)`
deltaS <- ranef(fit_lmer)$stat$`(Intercept)`
sd_deltaB <- as.numeric(arm::se.ranef(fit_lmer)$eq)
sd_deltaS <- as.numeric(arm::se.ranef(fit_lmer)$stat)
deltaWS <- data_reg$Y - predict(fit_lmer)
sd_deltaWS <- sqrt(sd_deltaB[eq]^2 + sd_deltaS[stat]^2) # approximately
##################
# fit using Inla
# priors for standard deviation paramters
prior_prec_tau <- list(prec = list(prior = 'pc.prec', param = c(0.3, 0.01)))
prior_prec_phiS2S <- list(prec = list(prior = 'pc.prec', param = c(0.3, 0.01)))
prior_prec_phiSS <- list(prec = list(prior = 'pc.prec', param = c(0.3, 0.01)))
form <- Y ~ M1 + M2 + MlogR + logR + R + Fss + Frv + logVS +
f(eq, model = "iid", hyper = prior_prec_tau) +
f(stat, model = "iid",hyper = prior_prec_phiS2S)
fit_inla <- inla(form,
data = data_reg,
family="gaussian",
control.family = list(hyper = prior_prec_phiSS),
quantiles = c(0.05,0.5,0.95)
)
sd_deltaS_inla <-fit_inla$summary.random$stat$sd
sd_deltaB_inla <-fit_inla$summary.random$eq$sd
sd_deltaWS_inla <- fit_inla$summary.fitted.values$sd
##################
# fit using Stan
mod <- cmdstan_model(file.path('./Git/MixedModels_Biases/', 'stan', 'gmm_full_qr.stan'))
data_list <- list(
N = n_rec,
NEQ = n_eq,
NSTAT = n_stat,
K = 9,
Y = as.numeric(data_reg$Y),
X = data_reg[,c("M1", "M2", "MlogR", "logR", "R", "Fss", "Frv", "logVS")], # design matrix
eq = eq,
stat = stat,
alpha = c(1,1,1)
)
fit_stan <- mod$sample(
data = data_list,
seed = 8472,
chains = 4,
iter_sampling = 200,
iter_warmup = 200,
refresh = 100,
max_treedepth = 10,
adapt_delta = 0.8,
parallel_chains = 2,
show_exceptions = FALSE
)## Running MCMC with 4 chains, at most 2 in parallel...
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##
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## Mean chain execution time: 29.3 seconds.
## Total execution time: 61.7 seconds.draws <- fit_stan$draws()
sd_deltaB_stan <- colSds(subset(as_draws_matrix(draws), variable = 'eqterm'))
sd_deltaS_stan <- colSds(subset(as_draws_matrix(draws), variable = 'statterm'))
sd_deltaWS_stan <- colSds(subset(as_draws_matrix(draws), variable = 'resid'))The estimated standard deviations are very similar between the three methods (we have used relatively weak priors, so their influence is not very strong). For Inla, we transform the mean estimate of the precisions into an estimate of the standard deviations, which is technicaly not correct, but for the sake of smplicity we keep it. The difference is small. Note that for the Bayesian models (Inla and Stan), the full output is not just a point estimate of \(\phi_{SS}\), \(\tau\), and \(\phi_{S2S}\), but the full posterior distribution.
df <- data.frame(inla = 1/sqrt(fit_inla$summary.hyperpar$mean),
lmer = c(phi_ss_lmer, tau_lmer, phi_s2s_lmer),
stan = colMeans(subset(as_draws_matrix(draws), variable = c('phi_ss','tau','phi_s2s'))),
row.names = c('phi_ss','tau','phi_s2s'))
knitr::kable(df, digits = 5, row.names = TRUE,
caption = "Comparison of standard deviation estimates.")| inla | lmer | stan | |
|---|---|---|---|
| phi_ss | 0.20407 | 0.20413 | 0.20415 |
| tau | 0.14141 | 0.14327 | 0.14436 |
| phi_s2s | 0.23341 | 0.23365 | 0.23314 |
Now we compare credible intervals (for the Bayesian models) and
confidence intervals (for lmer). For lmer, we
calculate the profile confidence intervals, using function
confint, while for the Bayesian credible intervals we take
the 5% and 95% quantile of the posterior distribution. Since
inla internally uses precisions for the scale parameter, we
convert the quantiles (precision to standard devation is a monotonic
transformation, so the quantiles do not change). We note that the
interpretation of confidence intervals can be tricky (Morey et al.
2016). The confidence and credible intervals agree quite
well.
## Computing profile confidence intervals ...df <- data.frame(inla_q05 = 1/sqrt(fit_inla$summary.hyperpar[,'0.95quant']),
inla_q95 = 1/sqrt(fit_inla$summary.hyperpar[,'0.05quant']),
lmer_c05 = ci_lmer[c(3,2,1),1],
lmer_c95 = ci_lmer[c(3,2,1),2],
stan_q05 = colQuantiles(subset(as_draws_matrix(draws), variable = c('phi_ss','tau','phi_s2s')),
probs = 0.05),
stan_q95 = colQuantiles(subset(as_draws_matrix(draws), variable = c('phi_ss','tau','phi_s2s')),
probs = 0.95),
row.names = c('phi_ss','tau','phi_s2s'))
knitr::kable(df, digits = 5, row.names = TRUE,
caption = "Comparison of standard deviation credible/confidence intervals.")| inla_q05 | inla_q95 | lmer_c05 | lmer_c95 | stan_q05 | stan_q95 | |
|---|---|---|---|---|---|---|
| phi_ss | 0.20023 | 0.20800 | 0.20025 | 0.20798 | 0.20041 | 0.20800 |
| tau | 0.12580 | 0.16383 | 0.12404 | 0.15880 | 0.12719 | 0.16362 |
| phi_s2s | 0.22163 | 0.24542 | 0.22216 | 0.24544 | 0.22238 | 0.24503 |
The coefficient estimates are also very close.
df <- data.frame(inla = fit_inla$summary.fixed$mean,
lmer = fixef(fit_lmer),
stan = colMeans(subset(as_draws_matrix(draws), variable = c('^c\\['), regex = TRUE)))
knitr::kable(df, digits = 5, row.names = TRUE,
caption = "Comparison of coefficient estimates.")| inla | lmer | stan | |
|---|---|---|---|
| (Intercept) | 3.40873 | 3.40922 | 3.40668 |
| M1 | 0.20329 | 0.20343 | 0.20201 |
| M2 | 0.00275 | 0.00256 | 0.01465 |
| MlogR | 0.28754 | 0.28764 | 0.28749 |
| logR | -1.39877 | -1.39899 | -1.39601 |
| R | -0.00309 | -0.00309 | -0.00311 |
| Fss | 0.11563 | 0.11583 | 0.11048 |
| Frv | -0.00142 | -0.00107 | -0.00363 |
| logVS | -0.42206 | -0.42193 | -0.42216 |
As are the confidence and credible intervals of the fixed effects.
df <- data.frame(inla_q05 = fit_inla$summary.fixed[,'0.05quant'],
inla_q95 = fit_inla$summary.fixed[,'0.95quant'],
lmer_c05 = ci_lmer[4:12,1],
lmer_c95 = ci_lmer[4:12,2],
stan_q05 = colQuantiles(subset(as_draws_matrix(draws), variable = c('^c\\['), regex = TRUE),
probs = 0.05),
stan_q95 = colQuantiles(subset(as_draws_matrix(draws), variable = c('^c\\['), regex = TRUE),
probs = 0.95))
knitr::kable(df, digits = 5, row.names = TRUE,
caption = "Comparison of coefficient credible/confidence intervals.")| inla_q05 | inla_q95 | lmer_c05 | lmer_c95 | stan_q05 | stan_q95 | |
|---|---|---|---|---|---|---|
| (Intercept) | 3.32703 | 3.49060 | 3.32740 | 3.49023 | 3.33255 | 3.48558 |
| M1 | 0.13983 | 0.26688 | 0.14014 | 0.26647 | 0.14216 | 0.26619 |
| M2 | -0.11604 | 0.12140 | -0.11532 | 0.12070 | -0.09652 | 0.12896 |
| MlogR | 0.26484 | 0.31024 | 0.26484 | 0.31019 | 0.26520 | 0.31024 |
| logR | -1.44813 | -1.34942 | -1.44810 | -1.34952 | -1.44679 | -1.34740 |
| R | -0.00342 | -0.00275 | -0.00342 | -0.00275 | -0.00343 | -0.00278 |
| Fss | 0.05622 | 0.17525 | 0.05652 | 0.17486 | 0.04949 | 0.17084 |
| Frv | -0.05706 | 0.05456 | -0.05676 | 0.05421 | -0.05467 | 0.04708 |
| logVS | -0.49610 | -0.34807 | -0.49600 | -0.34820 | -0.49422 | -0.34680 |
Below we plot the standard deviations of the random effects
(conditional standard deviations for lmer, standard
deviations of the posterior distribution for the Bayesian models)
against number of records per event/station. In general, they are
similar between all three fits. We see larger standard deviations of the
event terms for the Bayesian models for large magnitudes, which is due
to the fact that uncertainty due to coefficients is included, which is
larger for large magnitudes.
p1 <- data.frame(unique(data_it[,c('EQID','mag')]),
lmer = sd_deltaB,
inla = sd_deltaB_inla,
stan = sd_deltaB_stan,
nrec = as.numeric(table(eq))) |>
pivot_longer(c(lmer, inla, stan)) %>%
ggplot() +
geom_point(aes(x = nrec, y = value, color = name, size = mag)) +
scale_x_log10(breaks = breaks, minor_breaks = minor_breaks) +
ylim(c(0,0.125)) +
labs(x = 'number of records per event', y = TeX("$\\psi(\\widehat{\\delta B})$")) +
guides(color = guide_legend(title=NULL), size = guide_legend(title = 'M')) +
theme(legend.position = c(0.85,0.75))## Warning: A numeric `legend.position` argument in `theme()` was deprecated in ggplot2
## 3.5.0.
## ℹ Please use the `legend.position.inside` argument of `theme()` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.p2 <- data.frame(unique(data_it[,c('STATID','vs30')]),
logvs = unique(data_reg[,c('stat','logVS')])[,2],
lmer = sd_deltaS,
inla = sd_deltaS_inla,
stan = sd_deltaS_stan,
nrec = as.numeric(table(stat))) |>
pivot_longer(c(lmer, inla, stan)) %>%
ggplot() +
geom_point(aes(x = nrec, y = value, color = name, size = vs30)) +
scale_x_log10(breaks = breaks, minor_breaks = minor_breaks) +
ylim(c(0,0.18)) +
labs(x = 'number of records per station', y = TeX("$\\psi(\\widehat{\\delta S})$")) +
guides(color = guide_legend(title=NULL), size = guide_legend(title = 'VS30')) +
theme(legend.position = c(0.85,0.75))
patchwork::wrap_plots(p1,p2)## Warning: Removed 1 row containing missing values or values outside the scale range
## (`geom_point()`).
Below we
compare the standard deviations for \(\delta
WS\). For lmer, this is an approximation, calculated
as \(\psi(\widehat{\delta WS})^2 =
\psi(\widehat{\delta B})^2 + \psi(\widehat{\delta S})^2\). At
larger values of \(\psi(\widehat{\delta
WS})\), the values from lmer are larger compared to
the ones from Inla or Stan, which is probably due to some correlations
between (estimated) event terms and site terms. These corelations are
implicitly taken into account in the Bayesian models.
data.frame(inla = sd_deltaWS_inla,
lmer = sd_deltaWS,
stan = sd_deltaWS_stan) %>%
pivot_longer(!inla) %>%
ggplot() +
geom_point(aes(x = inla, y = value, color = name), size = 4) +
geom_abline(color = 'black', linewidth = 1.5) +
lims(x = c(0.03,0.21), y = c(0.03, 0.21)) +
labs(x = TeX("$\\psi(\\widehat{\\delta WS}_{inla})$"),
y = TeX("$\\psi(\\widehat{\\delta WS})$")) +
guides(color = guide_legend(title=NULL))
We now calculate the conditional standard deviations of the random
effects from the lmer fit according to Equation (3) of the
paper, which makes use of the design matrix of the random effects \(\mathbf{Z}\), and the relative covariance
factor \(\mathbf{\Lambda}\). We compare
the results against the values calculated by package arm,
and find that the differences are negligible.
Z <- getME(fit_lmer, 'Z') #sparse Z design matrix
lambda <- getME(fit_lmer, 'Lambda')
V <- sigma(fit_lmer)^2 * lambda %*% solve((t(lambda) %*% t(Z) %*% Z %*% lambda + diag(n_eq + n_stat))) %*% lambda
# station entries are first
print(c(sum((arm::se.ranef(fit_lmer)$stat)^2 - diag(V)[1:n_stat]),
sum((arm::se.ranef(fit_lmer)$eq)^2 - diag(V)[(n_stat + 1):(n_eq + n_stat)])))## [1] 7.350891e-16 2.257309e-161.3 Building Intution
As stated before, the estimated value of the standard deviation is the sum of the variance of the point estimates plus the average conditional variance. It can be tempting to think that the second term, which is the average uncertainty of the random effect, is related to the uncertainty of the estimate of the standard deviation, but that is not the case. To illustrate this point, we simulate some data for a simple radom effects model with one grouping variable. We look at two cases, one wherewe have many groups, but few observations per group, and one where we have a small number of groups, but each with many observations.
set.seed(1701)
sigma_gr <- 0.5 # group-level standard devation
sigma <- 0.7 # noise standard deviation
n_gr <- 500 # number of groups
n_rep <- 3 # number of observations in each group
gr <- rep(1:n_gr, each = n_rep) # group indicator
# sample data from normal distributions and combine
df_sim <- data.frame(y_sim = rnorm(n_gr, sd = sigma_gr)[gr] + rnorm(length(gr), sd = sigma),
gr= gr)
# fit random effects model and calculate confidence interval
fit_sim <- lmer(y_sim ~ (1 | gr), df_sim) # fit model
ci1 <- confint(fit_sim, level = 0.9)## Computing profile confidence intervals ...# extract random effects and uncertainties (conditional mode and conditional standard deviation)
# all conditional standard deviations should be the same
tmp <- as.data.frame(ranef(fit_sim))
dG <- tmp[tmp$grpvar == 'gr','condval']
sd_dG <- tmp[tmp$grpvar == 'gr','condsd']
# repeat the exercise, but with a different number ofgroups and observations per group
n_gr2 <- 15
n_rep2 <- 100
gr2 <- rep(1:n_gr2, each = n_rep2)
df_sim2 <- data.frame(y_sim = rnorm(n_gr2, sd = sigma_gr)[gr2] + rnorm(length(gr2), sd = sigma),
gr= gr2)
n_sim2 <- nrow(df_sim2)
fit_sim2 <- lmer(y_sim ~ (1 | gr), df_sim2)
ci2 <- confint(fit_sim2, level = 0.9)## Computing profile confidence intervals ...tmp <- as.data.frame(ranef(fit_sim2))
dG2 <- tmp[tmp$grpvar == 'gr','condval']
sd_dG2 <- tmp[tmp$grpvar == 'gr','condsd']
print(VarCorr(fit_sim))## Groups Name Std.Dev.
## gr (Intercept) 0.50489
## Residual 0.73133## Groups Name Std.Dev.
## gr (Intercept) 0.62455
## Residual 0.68885As we can see, for the first case the standard deviations are well estimated, while the group-level standard deviation in the second case is off. The reason is that we only have 15 groups, which leads to a very uncertain estimate. This is reflected in the confidence interval, shown below, which is very wide for the second case. On the other hand, the average conditional variance is small, due to the fact that we have many observations per group.
df = data.frame(cbind(rowDiffs(ci1), rowDiffs(ci2),
c(sum(sd_dG^2)/n_gr, NA, NA),
c(sum(sd_dG2^2)/n_gr2, NA, NA))) %>%
set_names(c('X95_1','X95_2','avg_var_1','avg_var_2'))
knitr::kable(df, digits = 5, row.names = TRUE,
caption = "90% confidence intervals for estimated parameters, for case 1 (large number of groups)
and case 2 (small number of groups).")| X95_1 | X95_2 | avg_var_1 | avg_var_2 | |
|---|---|---|---|---|
| .sig01 | 0.09322 | 0.38779 | 0.10491 | 0.00469 |
| .sigma | 0.05384 | 0.04161 | NA | NA |
| (Intercept) | 0.09687 | 0.53976 | NA | NA |
Nevertheless, the uncertainty about the values of the random effets leads to uncertainty about the value of the standard deviation. We can sample from the conditional distribution of the random effects, and calculate the standard deviation of the sampled values. If we repeat his multiple times, we get multiple values of the standard devation, and we could imagine that we an use these samples to get some sort of uncertainty estmate of the standard deviation.
Below, we sample 10,000 different standard deviations, calculate the
5% and 95% quantile of the sampled values, and compare it to the values
calculated with confint. The values calculated from
sampling are much narrower than the ones calculated with
confint. In the paper we show that the values calculated
with confint are well calibrated and provide a good
assessment of uncertainty.
n_rep <- 10000
sample_sd <- rep(NA, n_rep)
sample_sd2 <- rep(NA, n_rep)
for(i in 1:n_rep) {
sample_sd[i] <- sd(rnorm(n_gr, mean = dG, sd = sd_dG))
sample_sd2[i] <- sd(rnorm(n_gr2, mean = dG2, sd = sd_dG2))
}
df <- data.frame(rbind(c(ci1[1,], ci2[1,]),
c(quantile(sample_sd, probs = c(0.05,0.95)), quantile(sample_sd2, probs = c(0.05,0.95)))
)) %>% set_names('q0.05_1','q0.95_1','q0.05_2','q0.95_2')
row.names(df) <- c('confint','sample')
knitr::kable(df, digits = 5, row.names = TRUE,
caption = "Comparison of uncertainty from confint and sampling.")| q0.05_1 | q0.95_1 | q0.05_2 | q0.95_2 | |
|---|---|---|---|---|
| confint | 0.45808 | 0.55130 | 0.45683 | 0.84462 |
| sample | 0.48355 | 0.52637 | 0.59398 | 0.65403 |
Blow, we shw plots of the densities of the sampled standard
deviations, together wth the true (solid) value of
sigma_gr, the value estimated by lmer
(dashed), and the standard deviations of the conditional modes of the
random effects. For both data sets, the standard deviation from the
conditional modes is underestimatng the value estimated by
lmer, but is much closer for the second data set. Ths is
due to the large number of observations per group in this case. For both
data sets, the sampled standard deviations lie around the value from
lmer. We also show the 90% confidence interval from
confint in red. These intervals are much wider than the
sampled values.
p1 <- data.frame(sd = sample_sd) %>%
ggplot() +
geom_density(aes(x=sd), linewidth = lw) +
geom_vline(xintercept = c(sigma_gr, as.data.frame(VarCorr(fit_sim))$sdcor[1], sd(dG)),
linetype = c('solid','dashed','dotted'), linewidth = lw) +
geom_vline(xintercept = ci1[1,], color = 'red', linewidth = lw) +
labs(title = 'data 1', x = 'sigma_gr')
p2 <- data.frame(sd = sample_sd2) %>%
ggplot() +
geom_density(aes(x=sd), linewidth = lw) +
geom_vline(xintercept = c(sigma_gr, as.data.frame(VarCorr(fit_sim2))$sdcor[1], sd(dG2)),
linetype = c('solid','dashed','dotted'), linewidth = lw) +
geom_vline(xintercept = ci2[1,], color = 'red', linewidth = lw) +
labs(title = 'data 2', x = 'sigma_gr')
patchwork::wrap_plots(p1, p2)
2 Simulations using CB14 Data
We now turn to simulations to show how biases can occur when the uncertainty of random effects (event and site terms) as well as residuals is neglected. We first focus on standard deviations, which are underestimated when estimated from point estimates. This becomes a problem when standard deviations are modeled as heteroscedastic, e.e. dependent on predictor variables such as magnitude and/or distance.
We illustrate the underestimation of standard deviations on the WUS data from the GMM of Campbell and Bozorgnia (2014). In total, there are 12482 records from 274 events and 1519 stations.
data_reg <- read.csv(file.path('./Git/MixedModels_Biases/','/data','data_cb.csv'))
print(dim(data_reg))## [1] 12482 11## intercept eqid statid M VS Rrup Rjb eq stat M_stat VS_gmean
## 1 1 7 133 6.60 219.31 91.22 91.15 1 1 6.600 219.31
## 2 1 12 326 7.36 316.46 117.75 114.62 2 2 6.656 316.46
## 3 1 25 131 6.19 408.93 17.64 17.64 3 3 6.190 408.93
## 4 1 25 127 6.19 289.56 9.58 9.58 3 4 6.190 289.56
## 5 1 25 129 6.19 256.82 12.90 12.90 3 5 6.190 256.82
## 6 1 25 162 6.19 527.92 15.96 15.96 3 6 6.190 527.92p1 <- ggplot(data_reg) +
geom_point(aes(x = Rrup, y = M)) +
scale_x_log10(breaks = breaks, minor_breaks = minor_breaks)
p2 <- ggplot(unique(data_reg[,c('eqid','M')])) +
geom_histogram(aes(x = M))
patchwork::wrap_plots(p1, p2)## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
n_rec <- nrow(data_reg)
n_eq <- max(data_reg$eq)
n_stat <- max(data_reg$stat)
eq <- data_reg$eq
stat <- data_reg$stat
mageq <- unique(data_reg[,c('eq','M')])$M # event-specific magnitude
magstat <- unique(data_reg[,c('stat','M_stat')])$M_stat # station-specific magnitude
print(paste0('Number of records: ',n_rec,'; number of events: ',n_eq,'; number of stations: ',n_stat))## [1] "Number of records: 12482; number of events: 274; number of stations: 1519"2.1 Homoscedastic Standard Deviations
First, we simulate data using standard deviations that do not depend on any predictor variables, i.e. are homoscedastic. We do not simulate any fixed effects structure, in this example we focus on biases in the estimation of standard deviations.
First, we fix the standard deviations:
Next, we randomly sample event terms, site terms, and withn-event/within-site residuals, and combine them into total residuals (our target variable for this example).
set.seed(5618)
# randomly sample residals, event and site terms
dWS_sim <- rnorm(n_rec, sd = phi_ss_sim)
dS_sim <- rnorm(n_stat, sd = phi_s2s_sim)
dB_sim <- rnorm(n_eq, sd = tau_sim)
# combine into total residual/target variable
data_reg$y_sim <- dB_sim[eq] + dS_sim[stat] + dWS_simNow we perform the linear mixed effects regression using
lmer. We use maximum likelihood instead of restricted
maximum likelihood in this case to show the equivalence of the
calculations of standard deviations. In the paper, we also use Stan to
estimate the random effects and standard deviations, but we omit this
here to save time and space. As we have seen for the Italian data, we
get very similar results using lmer and Stan, and this is
also reflected by the results shown in the paper. Frequentist methods
are still overwhelmingly used in GMM development, so it makes sense to
focus on them here.
## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: y_sim ~ (1 | eq) + (1 | stat)
## Data: data_reg
##
## AIC BIC logLik deviance df.resid
## 20896.2 20925.9 -10444.1 20888.2 12478
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.6189 -0.6421 0.0026 0.6345 3.7999
##
## Random effects:
## Groups Name Variance Std.Dev.
## stat (Intercept) 0.1940 0.4404
## eq (Intercept) 0.1685 0.4105
## Residual 0.2473 0.4973
## Number of obs: 12482, groups: stat, 1519; eq, 274
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) -0.03459 0.03011 -1.149As we can see, the model parameters are quite well estimated. The intercept is close to zero, and the standard deviations are close to the values used in the simulation.
Below, we extract the conditional modes and standard deviations of the random effects. We also calculate the within-event/within-site residuals, and approximate their standard deviation.
tmp <- as.data.frame(ranef(fit_sim))
dS_lmer <- tmp[tmp$grpvar == 'stat','condval']
dB_lmer <- tmp[tmp$grpvar == 'eq','condval']
dWS_lmer <- data_reg$y_sim - predict(fit_sim)
sd_dS_lmer <- tmp[tmp$grpvar == 'stat','condsd']
sd_dB_lmer <- tmp[tmp$grpvar == 'eq','condsd']
sd_dWS_lmer <- sqrt(sd_dB_lmer[eq]^2 + sd_dS_lmer[stat]^2) # approximately
# alternative way to extract the random effects and conditional standard deviations
# dS_lmer <- ranef(fit_sim)$stat$`(Intercept)`
# dB_lmer <- ranef(fit_sim)$eq$`(Intercept)`
#
# sd_dB_lmer <- as.numeric(arm::se.ranef(fit_sim)$eq)
# sd_dS_lmer <- as.numeric(arm::se.ranef(fit_sim)$stat)Next, we compare different standard deviations. For all terms, we
show the true value used in the simulations, the standard devation of
the sampled terms, and then the value from the fit using
lmer. Then, we calculate the standard deviations according
to Equation (4) in the paper (including uncertainty). We also show the
standard deviations calculated based on the conditional modes of the
random effects/residuals, as well as calculated using a sample from the
conditional distribution.
# compare estiamtes of standard deviations
df <- data.frame(phi_s2s = c(phi_s2s_sim,
sd(dS_sim),
as.data.frame(VarCorr(fit_sim))$sdcor[1],
sqrt(sum(dS_lmer^2)/n_stat + sum(sd_dS_lmer^2)/n_stat),
sd(dS_lmer),
sd(rnorm(n_stat, mean = dS_lmer, sd = sd_dS_lmer))),
tau = c(tau_sim,
sd(dB_sim),
as.data.frame(VarCorr(fit_sim))$sdcor[2],
sqrt(sum(dB_lmer^2)/n_eq + sum(sd_dB_lmer^2)/n_eq),
sd(dB_lmer),
sd(rnorm(n_eq, mean = dB_lmer, sd = sd_dB_lmer))),
phi_ss = c(phi_ss_sim,
sd(dWS_sim),
as.data.frame(VarCorr(fit_sim))$sdcor[3],
sqrt(sum(dWS_lmer^2)/n_rec + sum(sd_dWS_lmer^2)/n_rec),
sd(dWS_lmer),
sd(rnorm(n_rec, mean = dWS_lmer, sd = sd_dWS_lmer))),
row.names = c('sim','sd(true)', 'lmer', 'with unc','sd(point estimate)','sd(sample)')
)
knitr::kable(df, digits = 5, row.names = TRUE,
caption = "Comparison of standard deviation estimates.")| phi_s2s | tau | phi_ss | |
|---|---|---|---|
| sim | 0.43000 | 0.40000 | 0.50000 |
| sd(true) | 0.43363 | 0.40595 | 0.49964 |
| lmer | 0.44042 | 0.41048 | 0.49731 |
| with unc | 0.44042 | 0.41048 | 0.49894 |
| sd(point estimate) | 0.35544 | 0.38813 | 0.47214 |
| sd(sample) | 0.44312 | 0.41292 | 0.50019 |
As we can see, the values from lmer and the ones
calculated according to Equation (4) agree for \(\tau\) and \(\phi_{S2S}\). For \(\phi_{SS}\), there is a small discrepancy,
since the conditional standard deviations are just an approximation.
These values are also close to the true ones, while the standard
deviations calculated from the point estimates are underestimating the
true values. The differences is largest for \(\phi_{S2S}\), since there are several
stations with only few recordings and thus large conditional standard
deviations. Sampling from the conditional distrbution of the random
effects/standard deviations leads to values that are closer to the true
ones.
Since there are many stations with very few recordings, the value of \(\phi_{S2S}\) is severely underestimated when calculated from the point estimates of the site terms. Thus, we now test whether what happens if we only use stations with at least 5 or 10 recordings. As we can see from the histogram (which shows 200 repeated simulations), on average the values are closer to the true value, but some bias remains. If one chooses to go this route, one also has to account for the fact that the estimates are based on fewer stations.
n_sam <- 200
res_s2s <- matrix(nrow = n_sam, ncol = 4)
set.seed(5618)
for(i in 1:n_sam) {
rect <- rnorm(n_rec, sd = phi_ss_sim)
statt <- rnorm(n_stat, sd = phi_s2s_sim)
eqtt <- rnorm(n_eq, sd = tau_sim)
data_reg$y_sim <- eqtt[eq] + statt[stat] + rect
fit_sim <- lmer(y_sim ~ (1 | eq) + (1 | stat), data_reg)
tmp <- ranef(fit_sim)$stat$`(Intercept)`
res_s2s[i,] <- c(as.data.frame(VarCorr(fit_sim))$sdcor[1],
sd(tmp), sd(tmp[table(stat) >= 5]), sd(tmp[table(stat) >= 10]))
}
data.frame(res_s2s) |> set_names(c('lmer', 'all','N_rec >= 5','N_rec >= 10')) |>
pivot_longer(everything()) |>
ggplot() +
geom_density(aes(x = value, color = name),linewidth = 1.5, key_glyph = draw_key_path) +
geom_vline(xintercept = phi_s2s_sim, linewidth = 1.5) +
guides(color = guide_legend(title = NULL)) +
xlab('phi_S2S') +
theme(legend.position = c(0.4,0.8))
In GMM development, the standard deviations are often modeled as dependent on some predictor variables such as magnitude. Bayless and Abrahamson (2018) contains a magnitude-dependent \(\phi_{S2S}\), which is modeled using the mean magnitude of all records by station. Kotha et al. (2022) performed a Breusch-Pagan test (Breusch and Pagan 1979) for heteroscedasticity to test for magnitude dependence of \(\tau\) and \(\phi_{SS}\). Below, we calcuale the p-values for the simulated data (which we know is not heteroscedastic). The null hypothesis is that the data is homoscedastc, and a low p-value is the probability of observing data if the null hypothesis were true. Based on point estimates, one would conclude that site terms and within-event/within-site residuals are heteroscedastic. In this context, be aware of hypothesis tests (Wasserstein, Schirm, and Lazar 2019),(Amrhein, Greenland, and McShane 2019).
# calculate p-value of Breusch-Pagan test, testing for dependence on magnitude
df <- data.frame(
dS = c(lmtest::bptest(dS ~ M, data = data.frame(M = magstat, dS = dS_lmer))$p.value,
lmtest::bptest(dS ~ M, data = data.frame(M = magstat, dS = rnorm(n_stat, mean = dS_lmer, sd = sd_dS_lmer)))$p.value,
lmtest::bptest(dS ~ M, data = data.frame(M = magstat, dS = dS_sim))$p.value),
dB = c(lmtest::bptest(dB ~ M, data = data.frame(M = mageq, dB = dB_lmer))$p.value,
lmtest::bptest(dB ~ M, data = data.frame(M = mageq, dB = rnorm(n_eq, mean = dB_lmer, sd = sd_dB_lmer)))$p.value,
lmtest::bptest(dB ~ M, data = data.frame(M = mageq, dB = dB_sim))$p.value),
dWS = c(lmtest::bptest(dWS ~ M, data = data.frame(M = data_reg$M, dWS = dWS_lmer))$p.value,
lmtest::bptest(dWS ~ M, data = data.frame(M = data_reg$M, dWS = rnorm(n_rec, mean = dWS_lmer, sd = sd_dWS_lmer)))$p.value,
lmtest::bptest(dWS ~ M, data = data.frame(M = data_reg$M, dWS = dWS_sim))$p.value),
row.names = c('point estimate','sample','true'))
knitr::kable(df, digits = 5, row.names = TRUE,
caption = "P-values from Breusch-Pagan test.")| dS | dB | dWS | |
|---|---|---|---|
| point estimate | 0.00000 | 0.99921 | 0.0000 |
| sample | 0.80356 | 0.53500 | 0.6652 |
| true | 0.10186 | 0.98637 | 0.8881 |
Below, we calculate the standard deviations of the site terms \(\delta S\), the event terms \(\delta B\), and the residuals \(\delta WS\), for different magnitude bins.
We calculate the standard deviations for the simulated (true) values,
the point estimates from the lmer fit, and for the estmated
random effects/residuals including their uncertainty. The true value of
the standard deviation is shown as a horizontal black line.
We can decreasing values wth magnitude in the standard deviations estimated from the point estimates for \(\phi_{S2S}\) and \(\phi_{SS}\), while the values calculated from the true samples are more constant (as they should be). While one always needs to be careful with different numbers of events/records/stations within each bin, plots like these (wth patterns as in these plots) are often used to conclude that standard deviations (\(\phi_{S2S}\) and \(\phi_{SS}\) in this case) should be modeled as magnitude dependent, which in this case is not true. The standard deviations that include uncertainty of the random effects/residuals show a pattern that is closr to constant.
# mamgnitude break points for bins
magbins <- c(3,4,5,6,7,8)
# site terms
df_stat <- data.frame(M = magstat, dS_sim, dS_lmer, sd_dS_lmer) %>%
mutate(bin = cut(M, breaks = magbins, labels = FALSE)) %>%
group_by(bin) %>%
mutate(sd_sim = sd(dS_sim),
sd_lmer = sd(dS_lmer),
sd_lmer_unc = sqrt((sum(dS_lmer^2) + sum(sd_dS_lmer^2))/length(dS_lmer^2)),
meanM = mean(M)) %>%
arrange(M)
p1 <- cbind(unique(df_stat[,c('sd_sim','sd_lmer','sd_lmer_unc','meanM')]),
m1 = magbins[1:(length(magbins)-1)],
m2 = magbins[2:length(magbins)]) |>
pivot_longer(!c(m1,m2,meanM)) |>
ggplot() +
geom_segment(aes(x = m1, xend = m2, y=value, yend = value, color = name), linewidth = 1.5) +
scale_color_manual(values = c('red','orange','blue'),
labels = c('lmer','lmer with unc','sim')) +
geom_hline(yintercept = phi_s2s_sim, linewidth = 1.5) +
guides(color = guide_legend(title=NULL)) +
labs(x = 'M', y = 'phi_S2S') +
theme(legend.position = c(0.2,0.2))
# event terms
df_eq <- data.frame(M = mageq, dB_sim, dB_lmer, sd_dB_lmer) %>%
mutate(bin = cut(M, breaks = magbins, labels = FALSE)) %>%
group_by(bin) %>%
mutate(sd_sim = sd(dB_sim),
sd_lmer = sd(dB_lmer),
sd_lmer_unc = sqrt((sum(dB_lmer^2) + sum(sd_dB_lmer^2))/length(dB_lmer^2)),
meanM = mean(M)) %>%
arrange(M)
p2 <- cbind(unique(df_eq[,c('sd_sim','sd_lmer','sd_lmer_unc','meanM')]),
m1 = magbins[1:(length(magbins)-1)],
m2 = magbins[2:length(magbins)]) |>
pivot_longer(!c(m1,m2,meanM)) |>
ggplot() +
geom_segment(aes(x = m1, xend = m2, y=value, yend = value, color = name), linewidth = 1.5) +
scale_color_manual(values = c('red','orange','blue'),
labels = c('lmer','lmer with unc','sim')) +
geom_hline(yintercept = tau_sim, linewidth = 1.5) +
guides(color = guide_legend(title=NULL)) +
labs(x = 'M', y = 'tau') +
theme(legend.position = 'none')
# residuals
df_rec <- data.frame(M = data_reg$M, dWS_sim, dWS_lmer, sd_dWS_lmer) %>%
mutate(bin = cut(M, breaks = magbins, labels = FALSE)) %>%
group_by(bin) %>%
mutate(sd_sim = sd(dWS_sim),
sd_lmer = sd(dWS_lmer),
sd_lmer_unc = sqrt((sum(dWS_lmer^2) + sum(sd_dWS_lmer^2))/length(dWS_lmer^2)),
meanM = mean(M)) %>%
arrange(M)
p3 <- cbind(unique(df_rec[,c('sd_sim','sd_lmer','sd_lmer_unc','meanM')]),
m1 = magbins[1:(length(magbins)-1)],
m2 = magbins[2:length(magbins)]) |>
pivot_longer(!c(m1,m2,meanM)) |>
ggplot() +
geom_segment(aes(x = m1, xend = m2, y=value, yend = value, color = name), linewidth = 1.5) +
scale_color_manual(values = c('red','orange','blue'),
labels = c('lmer','lmer with unc','sim')) +
geom_hline(yintercept = phi_ss_sim, linewidth = 1.5) +
guides(color = guide_legend(title=NULL)) +
labs(x = 'M', y = 'phi_SS') +
theme(legend.position = 'none')
patchwork::wrap_plots(p1,p2,p3, ncol = 2)
2.2 Magnitude-Dependent Tau and Phi_SS
In this section, we estimate magnitude dependent standard deviations. We simulate data with magnitude dependent \(\tau\) and \(\phi_{SS}\). The dependence has the form \[ \tau(M) = \left\{ \begin{array}{ll} {\tau}_1 & M \leq M_1 \\ {\tau}_1 + ({\tau}_2 - {\tau}_1) \frac{M - M_1}{M_2 - M_1} & M_1 < M < M_2 \\ {\tau}_2 & M \geq M_2 \end{array} \right. \] with a form similar for \(\phi_{SS}\).
For this simulation, we also generate median predictions from fixed effects, in order to check how well the coefficients are estimated.
First, we declare the values of the standard deviations for the simulations.
# fix standard deviations and magnitude break points
phi_s2s_sim <- 0.43
tau_sim_val <- c(0.4,0.25)
phi_ss_sim_val <- c(0.55,0.4)
mb_tau <- c(5,6)
mb_phi <- c(4.5,5.5)
# define tau for each event
# define linear predictors
m1_eq <- 1 * (mageq < mb_tau[2]) - (mageq - mb_tau[1]) / (mb_tau[2] - mb_tau[1]) * (mageq > mb_tau[1] & mageq < mb_tau[2])
m2_eq <- 1 * (mageq >= mb_tau[2]) + (mageq - mb_tau[1]) / (mb_tau[2] - mb_tau[1]) * (mageq > mb_tau[1] & mageq < mb_tau[2])
tau_sim <- m1_eq * tau_sim_val[1] + m2_eq * tau_sim_val[2]
# define phi_ss for each record
# define linear predictors
m1_rec <- 1 * (data_reg$M < mb_phi[2]) - (data_reg$M - mb_phi[1]) / (mb_phi[2] - mb_phi[1]) * (data_reg$M > mb_phi[1] & data_reg$M < mb_phi[2])
m2_rec <- 1 * (data_reg$M >= mb_phi[2]) + (data_reg$M - mb_phi[1]) / (mb_phi[2] - mb_phi[1]) * (data_reg$M > mb_phi[1] & data_reg$M < mb_phi[2])
phi_ss_sim <- m1_rec * phi_ss_sim_val[1] + m2_rec * phi_ss_sim_val[2]Now, we declare the coefficients, which are taken from the ITA18 model of Lanzano et al. (2019). We also compute the linear predictors for the model.
coeffs <- c(3.421046409, 0.193954090, -0.021982777, 0.287149291, -1.405635476, -0.002911264, -0.394575970)
names_coeffs <- c("intercept", "M1", "M2", "MlogR", "logR", "R", "logVS")
# Set linear predictors
mh = 5.5
mref = 5.324
h = 6.924
data_reg$M1 <- (data_reg$M-mh)*(data_reg$M<=mh)
data_reg$M2 <- (data_reg$M-mh)*(data_reg$M>mh)
data_reg$MlogR <- (data_reg$M-mref)*log10(sqrt(data_reg$Rjb^2+h^2))
data_reg$logR <- log10(sqrt(data_reg$Rjb^2+h^2))
data_reg$R <- sqrt(data_reg$Rjb^2+h^2)
data_reg$logVS <- log10(data_reg$VS_gmean/800)*(data_reg$VS_gmean<=1500)+log10(1500/800)*(data_reg$VS_gmean>1500)Now, we randomly sample event terms, site terms, and residuals, and combine with median predictions.
set.seed(1701)
dB_sim <- rnorm(n_eq, sd = tau_sim)
dWS_sim <- rnorm(n_rec, sd = phi_ss_sim)
dS_sim <- rnorm(n_stat, sd = phi_s2s_sim)
data_reg$y_sim <- as.matrix(data_reg[,names_coeffs]) %*% coeffs + dB_sim[eq] + dS_sim[stat] + dWS_simFirs, we perform a linear mixed effects regression (which assumes homoscedastic standard deviations). In general, the coefficients are estimated well, but the standard deviations are off.
# linear mixed effects regression
fit_sim <- lmer(y_sim ~ M1 + M2 + MlogR + logR + R + logVS + (1 | eq) + (1 | stat), data_reg)
summary(fit_sim)## Linear mixed model fit by REML ['lmerMod']
## Formula: y_sim ~ M1 + M2 + MlogR + logR + R + logVS + (1 | eq) + (1 |
## stat)
## Data: data_reg
##
## REML criterion at convergence: 22112.8
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -4.1164 -0.6395 0.0005 0.6258 3.8848
##
## Random effects:
## Groups Name Variance Std.Dev.
## stat (Intercept) 0.1706 0.4130
## eq (Intercept) 0.1458 0.3819
## Residual 0.2786 0.5278
## Number of obs: 12482, groups: stat, 1519; eq, 274
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 3.6438563 0.0981049 37.142
## M1 0.2919727 0.0548058 5.327
## M2 -0.0849910 0.0971780 -0.875
## MlogR 0.2676451 0.0178760 14.972
## logR -1.4212048 0.0389614 -36.477
## R -0.0029312 0.0001654 -17.723
## logVS -0.3182135 0.0902805 -3.525
##
## Correlation of Fixed Effects:
## (Intr) M1 M2 MlogR logR R
## M1 0.755
## M2 -0.311 -0.248
## MlogR -0.328 -0.542 -0.385
## logR -0.584 -0.241 -0.178 0.470
## R 0.389 0.029 0.038 -0.119 -0.798
## logVS 0.303 0.029 0.029 -0.011 -0.021 -0.035# extract conditional modes and standard deviations of residuals/random effects
tmp <- as.data.frame(ranef(fit_sim))
dS_lmer <- tmp[tmp$grpvar == 'stat','condval']
dB_lmer <- tmp[tmp$grpvar == 'eq','condval']
dWS_lmer <- data_reg$y_sim - predict(fit_sim)
sd_dS_lmer <- tmp[tmp$grpvar == 'stat','condsd']
sd_dB_lmer <- tmp[tmp$grpvar == 'eq','condsd']
sd_dWS_lmer <- sqrt(sd_dB_lmer[eq]^2 + sd_dS_lmer[stat]^2) # approximately
# calculate total resduals
dR_lmer <- data_reg$y_sim - predict(fit_sim, re.form=NA)Next, we run a Stan model (Carpenter et al. 2017), https://mc-stan.org/, on the total residuals. In the Stan model, we can mdel the standard deviations to be magnitude dependent. Below, we compile the Stan model, and print out its code.
mod <- cmdstan_model(file.path('./Git/MixedModels_Biases/', 'stan', 'gmm_partition_tauM_phiM.stan'))
mod## /*********************************************
## ********************************************/
##
## data {
## int N; // number of records
## int NEQ; // number of earthquakes
## int NSTAT; // number of stations
##
## vector[NEQ] MEQ;
## vector[N] Y; // ln ground-motion value
##
## array[N] int<lower=1,upper=NEQ> eq; // event id (in numerical order from 1 to last)
## array[N] int<lower=1,upper=NSTAT> stat; // station id (in numerical order from 1 to last)
##
## vector[NEQ] M1_eq;
## vector[NEQ] M2_eq;
## vector[N] M1_rec;
## vector[N] M2_rec;
## }
##
## parameters {
## real ic;
##
## real<lower=0> tau_1;
## real<lower=0> tau_2;
##
## real<lower=0> phi_ss_1;
## real<lower=0> phi_ss_2;
##
## real<lower=0> phi_s2s;
##
## vector[NEQ] eqterm;
## vector[NSTAT] statterm;
## }
##
##
## model {
## ic ~ normal(0,0.1);
##
## phi_s2s ~ normal(0,0.5);
##
## tau_1 ~ normal(0,0.5);
## tau_2 ~ normal(0,0.5);
##
## phi_ss_1 ~ normal(0,0.5);
## phi_ss_2 ~ normal(0,0.5);
##
## vector[N] phi_ss = M1_rec * phi_ss_1 + M2_rec * phi_ss_2;
## vector[NEQ] tau = M1_eq * tau_1 + M2_eq * tau_2;
##
## eqterm ~ normal(0, tau);
## statterm ~ normal(0, phi_s2s);
##
## Y ~ normal(ic + eqterm[eq] + statterm[stat], phi_ss);
## }Now, we declare the data for Stan, and run the model. To keep running time low, we only run 200 warm-up and 200 sampling iterations.
data_list <- list(
N = n_rec,
NEQ = n_eq,
NSTAT = n_stat,
Y = as.numeric(dR_lmer),
eq = eq,
stat = stat,
MEQ = mageq,
M1_eq = m1_eq,
M2_eq = m2_eq,
M1_rec = m1_rec,
M2_rec = m2_rec
)
fit <- mod$sample(
data = data_list,
seed = 8472,
chains = 4,
iter_sampling = 200,
iter_warmup = 200,
refresh = 100,
max_treedepth = 10,
adapt_delta = 0.8,
parallel_chains = 2,
show_exceptions = FALSE
)## Running MCMC with 4 chains, at most 2 in parallel...
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## Chain 4 finished in 201.9 seconds.
##
## All 4 chains finished successfully.
## Mean chain execution time: 169.1 seconds.
## Total execution time: 341.5 seconds.## Processing csv files: /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpmoLBg2/gmm_partition_tauM_phiM-202310021107-1-0da5d7.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpmoLBg2/gmm_partition_tauM_phiM-202310021107-2-0da5d7.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpmoLBg2/gmm_partition_tauM_phiM-202310021107-3-0da5d7.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpmoLBg2/gmm_partition_tauM_phiM-202310021107-4-0da5d7.csv
##
## Checking sampler transitions treedepth.
## Treedepth satisfactory for all transitions.
##
## Checking sampler transitions for divergences.
## No divergent transitions found.
##
## Checking E-BFMI - sampler transitions HMC potential energy.
## E-BFMI satisfactory.
##
## Effective sample size satisfactory.
##
## Split R-hat values satisfactory all parameters.
##
## Processing complete, no problems detected.
## $status
## [1] 0
##
## $stdout
## [1] "Processing csv files: /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpmoLBg2/gmm_partition_tauM_phiM-202310021107-1-0da5d7.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpmoLBg2/gmm_partition_tauM_phiM-202310021107-2-0da5d7.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpmoLBg2/gmm_partition_tauM_phiM-202310021107-3-0da5d7.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpmoLBg2/gmm_partition_tauM_phiM-202310021107-4-0da5d7.csv\n\nChecking sampler transitions treedepth.\nTreedepth satisfactory for all transitions.\n\nChecking sampler transitions for divergences.\nNo divergent transitions found.\n\nChecking E-BFMI - sampler transitions HMC potential energy.\nE-BFMI satisfactory.\n\nEffective sample size satisfactory.\n\nSplit R-hat values satisfactory all parameters.\n\nProcessing complete, no problems detected.\n"
##
## $stderr
## [1] ""
##
## $timeout
## [1] FALSE## $num_divergent
## [1] 0 0 0 0
##
## $num_max_treedepth
## [1] 0 0 0 0
##
## $ebfmi
## [1] 0.7268607 0.9408649 0.7289387 1.1460655draws_part <- fit$draws()
summarise_draws(subset(draws_part, variable = c('ic','phi','tau'), regex = TRUE))## # A tibble: 6 × 10
## variable mean median sd mad q5 q95 rhat ess_bulk
## <chr> <num> <num> <num> <num> <num> <num> <num> <num>
## 1 ic -0.00596 -0.00479 0.0260 0.0266 -0.0484 0.0370 1.02 91.2
## 2 phi_ss_1 0.550 0.550 0.00418 0.00399 0.543 0.557 1.00 812.
## 3 phi_ss_2 0.395 0.395 0.00831 0.00839 0.381 0.409 1.00 893.
## 4 phi_s2s 0.431 0.431 0.0109 0.0115 0.414 0.450 1.00 594.
## 5 tau_1 0.391 0.390 0.0202 0.0187 0.357 0.426 0.999 1775.
## 6 tau_2 0.319 0.314 0.0520 0.0507 0.240 0.412 1.00 912.
## # ℹ 1 more variable: ess_tail <num>In general, the parameters are well estimated. There are not that many events for \(M \geq 6\), so the value of \(\tau_2\) is quite uncertain.
In the following, we run a Stan model which estimates coefficients and magitude-dependent standard deviations at the same time. To improve sampling, we use the QR-decomposition of the desgn matrix.
mod <- cmdstan_model(file.path('./Git/MixedModels_Biases/', 'stan', 'gmm_full_qr_tauM_phiM.stan'))
mod## /*********************************************
## ********************************************/
##
## data {
## int N; // number of records
## int NEQ; // number of earthquakes
## int NSTAT; // number of stations
## int K;
##
## vector[NEQ] MEQ;
## matrix[N, K-1] X;
## vector[N] Y; // ln ground-motion value
##
## array[N] int<lower=1,upper=NEQ> eq; // event id (in numerical order from 1 to last)
## array[N] int<lower=1,upper=NSTAT> stat; // station id (in numerical order from 1 to last)
##
## vector[NEQ] M1_eq;
## vector[NEQ] M2_eq;
## vector[N] M1_rec;
## vector[N] M2_rec;
## }
##
## transformed data {
## matrix[N, K-1] Q_ast = qr_thin_Q(X) * sqrt(N - 1);
## matrix[K-1, K-1] R_ast = qr_thin_R(X) / sqrt(N - 1);
## matrix[K-1, K-1] R_ast_inverse = inverse(R_ast);
## }
##
## parameters {
## vector[K-1] c_qr;
## real ic;
##
## real<lower=0> tau_1;
## real<lower=0> tau_2;
##
## real<lower=0> phi_ss_1;
## real<lower=0> phi_ss_2;
##
## real<lower=0> phi_s2s;
##
## vector[NEQ] eqterm;
## vector[NSTAT] statterm;
## }
##
## model {
## ic ~ normal(0,5);
## c_qr ~ std_normal();
##
## phi_s2s ~ normal(0,0.5);
##
## tau_1 ~ normal(0,0.5);
## tau_2 ~ normal(0,0.5);
##
## phi_ss_1 ~ normal(0,0.5);
## phi_ss_2 ~ normal(0,0.5);
##
## vector[N] phi_ss = M1_rec * phi_ss_1 + M2_rec * phi_ss_2;
## vector[NEQ] tau = M1_eq * tau_1 + M2_eq * tau_2;
##
## eqterm ~ normal(0, tau);
## statterm ~ normal(0, phi_s2s);
##
## Y ~ normal(ic + Q_ast * c_qr + eqterm[eq] + statterm[stat], phi_ss);
##
## }
##
## generated quantities {
## vector[K] c;
## c[1] = ic;
## c[2:K] = R_ast_inverse * c_qr;
## }data_list <- list(
N = n_rec,
NEQ = n_eq,
NSTAT = n_stat,
K = length(coeffs),
Y = as.numeric(data_reg$y_sim),
X = data_reg[,c("M1", "M2", "MlogR", "logR", "R", "logVS")], # design matrix
eq = eq,
stat = stat,
MEQ = mageq,
M1_eq = m1_eq,
M2_eq = m2_eq,
M1_rec = m1_rec,
M2_rec = m2_rec
)
fit <- mod$sample(
data = data_list,
seed = 8472,
chains = 4,
iter_sampling = 200,
iter_warmup = 200,
refresh = 100,
max_treedepth = 10,
adapt_delta = 0.8,
parallel_chains = 2,
show_exceptions = FALSE
)## Running MCMC with 4 chains, at most 2 in parallel...
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## Chain 4 finished in 327.4 seconds.
##
## All 4 chains finished successfully.
## Mean chain execution time: 325.0 seconds.
## Total execution time: 655.1 seconds.## Processing csv files: /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpmoLBg2/gmm_full_qr_tauM_phiM-202310021113-1-03f6bb.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpmoLBg2/gmm_full_qr_tauM_phiM-202310021113-2-03f6bb.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpmoLBg2/gmm_full_qr_tauM_phiM-202310021113-3-03f6bb.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpmoLBg2/gmm_full_qr_tauM_phiM-202310021113-4-03f6bb.csv
##
## Checking sampler transitions treedepth.
## Treedepth satisfactory for all transitions.
##
## Checking sampler transitions for divergences.
## No divergent transitions found.
##
## Checking E-BFMI - sampler transitions HMC potential energy.
## E-BFMI satisfactory.
##
## Effective sample size satisfactory.
##
## Split R-hat values satisfactory all parameters.
##
## Processing complete, no problems detected.
## $status
## [1] 0
##
## $stdout
## [1] "Processing csv files: /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpmoLBg2/gmm_full_qr_tauM_phiM-202310021113-1-03f6bb.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpmoLBg2/gmm_full_qr_tauM_phiM-202310021113-2-03f6bb.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpmoLBg2/gmm_full_qr_tauM_phiM-202310021113-3-03f6bb.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpmoLBg2/gmm_full_qr_tauM_phiM-202310021113-4-03f6bb.csv\n\nChecking sampler transitions treedepth.\nTreedepth satisfactory for all transitions.\n\nChecking sampler transitions for divergences.\nNo divergent transitions found.\n\nChecking E-BFMI - sampler transitions HMC potential energy.\nE-BFMI satisfactory.\n\nEffective sample size satisfactory.\n\nSplit R-hat values satisfactory all parameters.\n\nProcessing complete, no problems detected.\n"
##
## $stderr
## [1] ""
##
## $timeout
## [1] FALSE## $num_divergent
## [1] 0 0 0 0
##
## $num_max_treedepth
## [1] 0 0 0 0
##
## $ebfmi
## [1] 0.8592778 1.0110550 0.7789643 0.8587937draws_full <- fit$draws()
summarise_draws(subset(draws_full, variable = c('phi','tau'), regex = TRUE))## # A tibble: 5 × 10
## variable mean median sd mad q5 q95 rhat ess_bulk ess_tail
## <chr> <num> <num> <num> <num> <num> <num> <num> <num> <num>
## 1 phi_ss_1 0.550 0.550 0.00393 0.00397 0.544 0.557 1.01 957. 607.
## 2 phi_ss_2 0.395 0.395 0.00860 0.00822 0.380 0.409 1.00 661. 800.
## 3 phi_s2s 0.433 0.432 0.0110 0.0114 0.415 0.451 1.01 609. 674.
## 4 tau_1 0.392 0.391 0.0206 0.0192 0.358 0.430 0.999 1622. 600.
## 5 tau_2 0.322 0.314 0.0628 0.0582 0.235 0.436 1.01 601. 803.## # A tibble: 7 × 10
## variable mean median sd mad q5 q95 rhat ess_bulk
## <chr> <num> <num> <num> <num> <num> <num> <num> <num>
## 1 c[1] 3.60 3.60 0.0986 0.0947 3.44 3.77 1.04 112.
## 2 c[2] 0.274 0.274 0.0521 0.0511 0.188 0.357 1.04 119.
## 3 c[3] -0.0621 -0.0610 0.0885 0.0820 -0.206 0.0789 1.03 179.
## 4 c[4] 0.268 0.268 0.0156 0.0157 0.241 0.292 1.00 1239.
## 5 c[5] -1.42 -1.42 0.0377 0.0399 -1.47 -1.36 1.01 734.
## 6 c[6] -0.00295 -0.00295 0.000166 0.000171 -0.00323 -0.00268 1.01 998.
## 7 c[7] -0.336 -0.333 0.0886 0.0880 -0.489 -0.192 1.01 209.
## # ℹ 1 more variable: ess_tail <num>The standard deviations are well estimated (very similar to the
values based on partitioning the total residuals from the
lmer fit), and the coefficients are also well
estimated.
Below, we plot the posterior distribution of \(\tau_1\) and \(\tau_2\), together with the true value (black) and the value estimated from point estimates of the event terms in the respective magnitude bins (red), with (solid) and without (dashed) uncertainty.
tmp <- mageq <= mb_tau[1]
tmp <- sqrt((sum(dB_lmer[tmp]^2) + sum(sd_dB_lmer[tmp]^2)) / sum(tmp))
p1 <- data.frame(dR = subset(as_draws_matrix(draws_part), variable = 'tau_1', regex = FALSE),
full = subset(as_draws_matrix(draws_full), variable = 'tau_1', regex = FALSE)) |>
set_names(c('dR','full')) |>
pivot_longer(everything()) |>
ggplot() +
geom_density(aes(x = value, color = name), linewidth = 1.5, key_glyph = draw_key_path) +
geom_vline(xintercept = tau_sim_val[1], linewidth = 1.5) +
geom_vline(xintercept = sd(dB_lmer[mageq <= mb_tau[1]]), linewidth = 1.5, color = 'red') +
geom_vline(xintercept = tmp, , linewidth = 1.5, color = 'red', linetype = 'dashed') +
guides(color = guide_legend(title = NULL)) +
theme(legend.position = c(0.8,0.8)) +
xlab('tau_1')
tmp <- mageq >= mb_tau[2]
tmp <- sqrt((sum(dB_lmer[tmp]^2) + sum(sd_dB_lmer[tmp]^2)) / sum(tmp))
p2 <- data.frame(dR = subset(as_draws_matrix(draws_part), variable = 'tau_2', regex = FALSE),
full = subset(as_draws_matrix(draws_full), variable = 'tau_2', regex = FALSE)) |>
set_names(c('dR','full')) |>
pivot_longer(everything()) |>
ggplot() +
geom_density(aes(x = value, color = name), linewidth = 1.5, key_glyph = draw_key_path) +
geom_vline(xintercept = tau_sim_val[2], linewidth = 1.5) +
geom_vline(xintercept = sd(dB_lmer[mageq >= mb_tau[2]]), linewidth = 1.5, color = 'red') +
geom_vline(xintercept = tmp, , linewidth = 1.5, color = 'red', linetype = 'dashed') +
guides(color = guide_legend(title = NULL)) +
theme(legend.position = c(0.8,0.8)) +
xlab('tau_2')
patchwork::wrap_plots(p1,p2)
While we see here that the values of \(\tau_1\) and \(\tau_2\) are estimated ok from
lmer, in the paper we show results from 100 simulations
which reveal on average a strong bias, whereas estimates from the Stan
models are on average better.
Below, we show posterior distributions of \(\phi_{SS,1}\) and \(\phi_{SS,2}\), similar to the plots for
\(\tau_1\) and \(\tau_2\). In this case, we see strong
biases for the estimates from lmer.
tmp <- data_reg$M <= mb_phi[1]
tmp <- sqrt((sum(dWS_lmer[tmp]^2) + sum(sd_dWS_lmer[tmp]^2)) / sum(tmp))
p1 <- data.frame(dR = subset(as_draws_matrix(draws_part), variable = 'phi_ss_1', regex = FALSE),
full = subset(as_draws_matrix(draws_full), variable = 'phi_ss_1', regex = FALSE)) |>
set_names(c('dR','full')) |>
pivot_longer(everything()) |>
ggplot() +
geom_density(aes(x = value, color = name), linewidth = 1.5, key_glyph = draw_key_path) +
geom_vline(xintercept = phi_ss_sim_val[1], linewidth = 1.5) +
geom_vline(xintercept = sd(dWS_lmer[data_reg$M <= mb_phi[1]]), linewidth = 1.5, color = 'red') +
geom_vline(xintercept = tmp, linewidth = 1.5, color = 'red', linetype = 'dashed') +
guides(color = guide_legend(title = NULL)) +
theme(legend.position = c(0.8,0.8)) +
xlab('phi_ss_1')
tmp <- data_reg$M >= mb_phi[2]
tmp <- sqrt((sum(dWS_lmer[tmp]^2) + sum(sd_dWS_lmer[tmp]^2)) / sum(tmp))
p2 <- data.frame(dR = subset(as_draws_matrix(draws_part), variable = 'phi_ss_2', regex = FALSE),
full = subset(as_draws_matrix(draws_full), variable = 'phi_ss_2', regex = FALSE)) |>
set_names(c('dR','full')) |>
pivot_longer(everything()) |>
ggplot() +
geom_density(aes(x = value, color = name), linewidth = 1.5, key_glyph = draw_key_path) +
geom_vline(xintercept = phi_ss_sim_val[2], linewidth = 1.5) +
geom_vline(xintercept = sd(dWS_lmer[data_reg$M >= mb_phi[2]]), linewidth = 1.5, color = 'red') +
geom_vline(xintercept = tmp, linewidth = 1.5, color = 'red', linetype = 'dashed') +
guides(color = guide_legend(title = NULL)) +
theme(legend.position = c(0.8,0.8)) +
xlab('phi_ss_2')
patchwork::wrap_plots(p1,p2)
And finally, the posterior distribution of \(\phi_{S2S}\).
data.frame(dR = subset(as_draws_matrix(draws_part), variable = 'phi_s2s', regex = FALSE),
full = subset(as_draws_matrix(draws_full), variable = 'phi_s2s', regex = FALSE)) |>
set_names(c('dR','full')) |>
pivot_longer(everything()) |>
ggplot() +
geom_density(aes(x = value, color = name), linewidth = 1.5, key_glyph = draw_key_path) +
geom_vline(xintercept = phi_s2s_sim, linewidth = 1.5) +
geom_vline(xintercept = sd(dS_lmer), linewidth = 1.5, color = 'red') +
geom_vline(xintercept = sqrt((sum(dS_lmer^2) + sum(sd_dS_lmer^2)) / n_stat),
linewidth = 1.5, color = 'red', linetype = 'dashed') +
guides(color = guide_legend(title = NULL)) +
theme(legend.position = c(0.3,0.8)) +
xlab('phi_S2S')
We can conclude from this smulation (and the repeated ones in the paper) that the magnitude-dependent standard deviations can be estimated using Stan from total residuals, but one should also account for uncertainty. Estimating the values from binned random effects/residuals can work but leads to a larger bias.
Our focus is on estimating the magnitude-dependent standard
deviations, but as a check we also plot the posterior distrbutions of
the coefficients for the full stan model, together with the true values
(black) and lmer estimates (red).
df <- data.frame(Parameter = c('c[1]','c[2]','c[3]','c[4]','c[5]','c[6]','c[7]'),
true = coeffs, lmer = fixef(fit_sim))
mcmc_hist(draws_full,regex_pars = 'c\\[') +
geom_vline(aes(xintercept = true), data = df, linewidth = 1.5) +
geom_vline(aes(xintercept = lmer), data = df, linewidth = 1.5, color = 'red')## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
2.2.1 Estimating Break Points
In the previous section, we have simulated data with
magnitude-dependent \(\tau\) and \(\phi\), using a trilnear function for the
magnitude dependence. In the Stan code, we have used the same fuctional
form, and assumed that the magnitude break points are known. Here, we
relax this assumption. We cannot directly estimate the break points
since the Hamiltonian Monte Carlo algorithm of Stan relies on
differentiation the model with respect to the parameters (loosely
speaking), which leads to problems for functions with sharp beaks.
Instead, we model the dependence using the
`logistic sigmoid'' function (calledinv_logit` in Stan),
which is defined as \[
\sigma(x) = \frac{1}{1 + \exp(-x)}
\] Below, we run a Stan model where both \(\phi_{SS}\) and \(\tau\) are modeled using this function.
mod <- cmdstan_model(file.path('./Git/MixedModels_Biases/', 'stan', 'gmm_partition_tauM_phiM_invlogit.stan'))
mod## /*********************************************
## ********************************************/
##
## data {
## int N; // number of records
## int NEQ; // number of earthquakes
## int NSTAT; // number of stations
##
## vector[NEQ] MEQ;
## vector[N] Y; // ln ground-motion value
##
## array[N] int<lower=1,upper=NEQ> eq; // event id (in numerical order from 1 to last)
## array[N] int<lower=1,upper=NSTAT> stat; // station id (in numerical order from 1 to last)
##
## }
##
## parameters {
## real ic;
##
## real<lower=0> tau_1;
## real<upper=tau_1> tau_2;
## real<lower=0> tau_scale;
## real<lower=4> mb_tau;
##
## real<lower=0> phi_ss_1;
## real<upper=phi_ss_1> phi_ss_2;
## real<lower=0> phi_ss_scale;
## real<lower=4> mb_phi_ss;
##
## real<lower=0> phi_s2s;
##
## vector[NEQ] eqterm;
## vector[NSTAT] statterm;
## }
##
## transformed parameters {
## }
##
##
## model {
## ic ~ normal(0,0.1);
##
## phi_s2s ~ normal(0,0.5);
##
## tau_1 ~ normal(0,0.5);
## tau_2 ~ normal(0,0.5);
## mb_tau ~ normal(5.,1);
## tau_scale ~ normal(6,1);
##
## phi_ss_1 ~ normal(0,0.5);
## phi_ss_2 ~ normal(0,0.5);
## mb_phi_ss ~ normal(5.,1);
## phi_ss_scale ~ normal(6,1);
##
## vector[NEQ] tau = tau_1 - tau_2 * inv_logit(tau_scale * (MEQ - mb_tau));
## vector[N] phi_ss = phi_ss_1 - phi_ss_2 * inv_logit(phi_ss_scale * (MEQ[eq] - mb_phi_ss));
##
## eqterm ~ normal(0, tau);
## statterm ~ normal(0, phi_s2s);
##
## Y ~ normal(ic + eqterm[eq] + statterm[stat], phi_ss);
##
## }data_list <- list(
N = n_rec,
NEQ = n_eq,
NSTAT = n_stat,
Y = as.numeric(dR_lmer),
eq = eq,
stat = stat,
MEQ = mageq
)
fit <- mod$sample(
data = data_list,
seed = 8472,
chains = 4,
iter_sampling = 200,
iter_warmup = 200,
refresh = 100,
max_treedepth = 10,
adapt_delta = 0.8, # increase to avoid divergences
parallel_chains = 2,
show_exceptions = FALSE
)## Running MCMC with 4 chains, at most 2 in parallel...
##
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## Chain 3 finished in 135.4 seconds.
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## Chain 4 finished in 133.0 seconds.
##
## All 4 chains finished successfully.
## Mean chain execution time: 175.7 seconds.
## Total execution time: 359.3 seconds.## Processing csv files: /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpY2WJGx/gmm_partition_tauM_phiM_invlogit-202310021433-1-967859.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpY2WJGx/gmm_partition_tauM_phiM_invlogit-202310021433-2-967859.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpY2WJGx/gmm_partition_tauM_phiM_invlogit-202310021433-3-967859.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpY2WJGx/gmm_partition_tauM_phiM_invlogit-202310021433-4-967859.csv
##
## Checking sampler transitions treedepth.
## Treedepth satisfactory for all transitions.
##
## Checking sampler transitions for divergences.
## No divergent transitions found.
##
## Checking E-BFMI - sampler transitions HMC potential energy.
## E-BFMI satisfactory.
##
## Effective sample size satisfactory.
##
## The following parameters had split R-hat greater than 1.05:
## ic, eqterm[47], eqterm[54], eqterm[90], eqterm[104], eqterm[143], eqterm[165]
## Such high values indicate incomplete mixing and biased estimation.
## You should consider regularizating your model with additional prior information or a more effective parameterization.
##
## Processing complete.
## $status
## [1] 0
##
## $stdout
## [1] "Processing csv files: /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpY2WJGx/gmm_partition_tauM_phiM_invlogit-202310021433-1-967859.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpY2WJGx/gmm_partition_tauM_phiM_invlogit-202310021433-2-967859.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpY2WJGx/gmm_partition_tauM_phiM_invlogit-202310021433-3-967859.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpY2WJGx/gmm_partition_tauM_phiM_invlogit-202310021433-4-967859.csv\n\nChecking sampler transitions treedepth.\nTreedepth satisfactory for all transitions.\n\nChecking sampler transitions for divergences.\nNo divergent transitions found.\n\nChecking E-BFMI - sampler transitions HMC potential energy.\nE-BFMI satisfactory.\n\nEffective sample size satisfactory.\n\nThe following parameters had split R-hat greater than 1.05:\n ic, eqterm[47], eqterm[54], eqterm[90], eqterm[104], eqterm[143], eqterm[165]\nSuch high values indicate incomplete mixing and biased estimation.\nYou should consider regularizating your model with additional prior information or a more effective parameterization.\n\nProcessing complete.\n"
##
## $stderr
## [1] ""
##
## $timeout
## [1] FALSE## $num_divergent
## [1] 0 0 0 0
##
## $num_max_treedepth
## [1] 0 0 0 0
##
## $ebfmi
## [1] 0.8298975 0.9716006 0.8808843 1.0236464draws_part2 <- fit$draws()
summarise_draws(subset(draws_part2, variable = c('ic','phi','tau'), regex = TRUE))## # A tibble: 10 × 10
## variable mean median sd mad q5 q95 rhat ess_bulk
## <chr> <num> <num> <num> <num> <num> <num> <num> <num>
## 1 ic -0.00338 -0.00238 0.0298 0.0306 -0.0542 0.0409 1.11 30.9
## 2 phi_ss_1 0.551 0.551 0.00473 0.00443 0.543 0.559 1.00 1191.
## 3 phi_ss_2 0.152 0.152 0.00982 0.00968 0.136 0.168 1.01 692.
## 4 phi_ss_scale 6.40 6.39 0.873 0.850 5.01 7.88 1.01 1034.
## 5 mb_phi_ss 4.96 4.96 0.0595 0.0605 4.87 5.06 1.00 807.
## 6 phi_s2s 0.431 0.431 0.0117 0.0118 0.413 0.451 0.997 500.
## 7 tau_1 0.391 0.390 0.0210 0.0217 0.360 0.426 1.00 1481.
## 8 tau_2 0.0679 0.0630 0.0791 0.0692 -0.0578 0.212 1.01 407.
## 9 tau_scale 5.94 5.95 0.988 0.963 4.35 7.57 1.00 925.
## 10 mb_tau 5.66 5.57 0.810 0.838 4.42 7.03 1.00 449.
## # ℹ 1 more variable: ess_tail <num>We get some warnings about incomplete mixing, which we will ignore here. We ranonly 400 iterations in total, and some longer chains probably lead to R-hat values that are closer to 1.
We now plot the magnitude dependence of \(\tau\) and \(\phi_{SS}\), estimated using the correct functional form and the logistic sigmoid function. For \(\tau\), the end values are similiar, but the transition is not well estimated with the logistic sigmoid function. Ths s due to the fact that there are not many events in total (and not many at large magnitudes), which makes the estimation difficult. For \(\phi_{SS}\), on the other hand, the two estimated functions agree quite well.
# function to calculate lnear predictos for magnitude scaling with break points
func_sd_mag <- function(mag, mb) {
m1 <- 1 * (mag < mb[2]) - (mag - mb[1]) / (mb[2] - mb[1]) * (mag > mb[1] & mag < mb[2])
m2 <- 1 * (mag >= mb[2]) + (mag - mb[1]) / (mb[2] - mb[1]) * (mag > mb[1] & mag < mb[2])
return(list(m1 = m1, m2 = m2))
}
# logistic sigmoid function
logsig <- function(x) {1/(1 + exp(-x))}
# magnitudes for plotting
mags_f <- seq(3,8,by=0.1)
mags <- func_sd_mag(mags_f, mb_tau)
tau_m <- as_draws_matrix(subset(draws_part, variable = 'tau_1')) %*% matrix(mags$m1, nrow = 1) +
as_draws_matrix(subset(draws_part, variable = 'tau_2')) %*% matrix(mags$m2, nrow = 1)
# posterior distribution of tau values for different magnitudes
tau_m_sig <- sapply(mags_f,
function(m) {as_draws_matrix(subset(draws_part2, variable = 'tau_1')) -
as_draws_matrix(subset(draws_part2, variable = 'tau_2')) *
logsig((m - as_draws_matrix(subset(draws_part2, variable = 'mb_tau'))) *
as_draws_matrix(subset(draws_part2, variable = 'tau_scale')))})
p1 <- data.frame(M = mags_f,
mean_true = tau_sim_val[1] * mags$m1 + tau_sim_val[2] * mags$m2,
mean_mod1 = colMeans(tau_m),
q05_mod1 = colQuantiles(tau_m, probs = 0.05),
q95_mod1 = colQuantiles(tau_m, probs = 0.95),
mean_mod2 = colMeans(tau_m_sig),
q05_mod2 = colQuantiles(tau_m_sig, probs = 0.05),
q95_mod2 = colQuantiles(tau_m_sig, probs = 0.95)) |>
pivot_longer(!M, names_sep = '_', names_to = c('par','mod')) |>
ggplot() +
geom_line(aes(x = M, y=value, color = mod, linetype = par), linewidth = 1.5) +
guides(color = guide_legend(title = NULL), linetype = guide_legend(title = NULL)) +
theme(legend.position = c(0.2,0.3)) +
scale_color_manual(values = c('red','blue','black'),
labels = c('tri-linear','logistic sigmoid','true')) +
labs(x = 'M', y = 'tau')
############ phi
mags <- func_sd_mag(mags_f, mb_phi)
phi_m <- as_draws_matrix(subset(draws_part, variable = 'phi_ss_1')) %*% matrix(mags$m1, nrow = 1) +
as_draws_matrix(subset(draws_part, variable = 'phi_ss_2')) %*% matrix(mags$m2, nrow = 1)
# posterior distribution of tau values for different magnitudes
phi_m_sig <- sapply(mags_f,
function(m) {as_draws_matrix(subset(draws_part2, variable = 'phi_ss_1')) -
as_draws_matrix(subset(draws_part2, variable = 'phi_ss_2')) *
logsig((m - as_draws_matrix(subset(draws_part2, variable = 'mb_phi_ss'))) *
as_draws_matrix(subset(draws_part2, variable = 'phi_ss_scale')))})
p2 <- data.frame(M = mags_f,
mean_true = phi_ss_sim_val[1] * mags$m1 + phi_ss_sim_val[2] * mags$m2,
mean_mod1 = colMeans(phi_m),
q05_mod1 = colQuantiles(phi_m, probs = 0.05),
q95_mod1 = colQuantiles(phi_m, probs = 0.95),
mean_mod2 = colMeans(phi_m_sig),
q05_mod2 = colQuantiles(phi_m_sig, probs = 0.05),
q95_mod2 = colQuantiles(phi_m_sig, probs = 0.95)) |>
pivot_longer(!M, names_sep = '_', names_to = c('par','mod')) |>
ggplot() +
geom_line(aes(x = M, y=value, color = mod, linetype = par), linewidth = 1.5) +
guides(color = guide_legend(title = NULL), linetype = guide_legend(title = NULL)) +
theme(legend.position = c(0.2,0.3)) +
scale_color_manual(values = c('red','blue','black'),
labels = c('tri-linear','logistic sigmoid','true')) +
labs(x = 'M', y = 'phi_SS')
patchwork::wrap_plots(p1, p2)
2.3 Estimating Scaling from Point Estimates of Random Effects
Random effects are sometimes used to estimate scaling of ground motions wth respect to new parameters, such as parameters associated with horizontal-to-vertical ratios. To assess potential biases, we simulate synthetic data using the ITA18 functional form, and then estimate a model without \(V_{S30}\)-scaling. We then estimate the \(V_{S30}\)-scaling coefficient from site terms.
The coefficients and linear predictors are already set. Here, we use standard deviations in \(log_{10}\)-units, which is what was used in Lanzano et al. (2019). We generate some data, and fit a full model (including \(V_{S30}\)-scaling).
tau_sim <- 0.17
phi_s2s_sim <- 0.23
phi_sim <- 0.2
set.seed(1701)
dB_sim <- rnorm(n_eq, mean =0, sd = tau_sim)
dS_sim <- rnorm(n_stat, mean =0, sd = phi_s2s_sim)
dWS_sim <- rnorm(n_rec, mean = 0, sd = phi_sim)
data_reg$y_sim <- as.matrix(data_reg[,names_coeffs]) %*% coeffs + dB_sim[eq] + dS_sim[stat] + dWS_sim
fit_sim <- lmer(y_sim ~ M1 + M2 + MlogR + logR + R + logVS + (1|eq) + (1|stat), data_reg)
summary(fit_sim)## Linear mixed model fit by REML ['lmerMod']
## Formula: y_sim ~ M1 + M2 + MlogR + logR + R + logVS + (1 | eq) + (1 |
## stat)
## Data: data_reg
##
## REML criterion at convergence: -1280.9
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -4.0218 -0.6377 0.0026 0.6275 3.9526
##
## Random effects:
## Groups Name Variance Std.Dev.
## stat (Intercept) 0.05519 0.2349
## eq (Intercept) 0.02733 0.1653
## Residual 0.03971 0.1993
## Number of obs: 12482, groups: stat, 1519; eq, 274
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 3.478e+00 4.206e-02 82.685
## M1 2.049e-01 2.313e-02 8.862
## M2 -1.067e-01 4.156e-02 -2.567
## MlogR 2.945e-01 7.160e-03 41.131
## logR -1.399e+00 1.545e-02 -90.557
## R -2.948e-03 6.396e-05 -46.099
## logVS -4.171e-01 4.608e-02 -9.051
##
## Correlation of Fixed Effects:
## (Intr) M1 M2 MlogR logR R
## M1 0.760
## M2 -0.320 -0.275
## MlogR -0.330 -0.521 -0.363
## logR -0.547 -0.253 -0.182 0.506
## R 0.359 0.036 0.040 -0.134 -0.789
## logVS 0.357 0.028 0.031 -0.020 -0.022 -0.025Now, we fit the model without the logVS term, and then
use linear regression on the estimated station term to estiamte the
coefficient. To account for stations with few recordings, we also use
only estmated site terms from staions with at least 10 records. We also
fit linear mixed effects model on the total residuals.
fit_sim2 <- lmer(y_sim ~ M1 + M2 + MlogR + logR + R + (1|eq) +(1|stat), data_reg)
deltaS_lmer2 <- ranef(fit_sim2)$stat$`(Intercept)`
deltaB_lmer2 <- ranef(fit_sim2)$eq$`(Intercept)`
data_reg$deltaR_lmer2 <- data_reg$y_sim - predict(fit_sim2, re.form=NA)
tmp <- data.frame(unique(data_reg[,c('stat','logVS')]), deltaS_lmer2 = deltaS_lmer2)
### all station terms
fit_sim2a <- lm(deltaS_lmer2 ~ logVS, tmp)
# summary(fit_sim2a)
### subset station ters
idx <- as.numeric(which(table(stat) >= 10))
fit_sim3a <- lm(deltaS_lmer2 ~ logVS, tmp[idx,])
# summary(fit_sim3a)
### total residuals
fit_sim4 <- lmer(deltaR_lmer2 ~ logVS + (1|eq) + (1|stat), data_reg)
df <- data.frame(name = c('true','full','dS(all)','dS(N>=10)','dR'),
value = c(coeffs[7], fixef(fit_sim)[7], fit_sim2a$coefficients[2],
fit_sim3a$coefficients[2], fixef(fit_sim4)[2]))
knitr::kable(df, digits = 5, row.names = TRUE,
caption = "Estimated VS30-scaling coefficient")| name | value | |
|---|---|---|
| 1 | true | -0.39458 |
| 2 | full | -0.41709 |
| 3 | dS(all) | -0.30915 |
| 4 | dS(N>=10) | -0.34638 |
| 5 | dR | -0.41379 |
As we can see, the \(V_{S30}\)-scaling coefficient estimated from site terms \(\delta S\) is strongly biased. The bias persists even if only stations with 10 or more recordings are used. A mixed-effects model on the total residuals works well to estimate the coefficient.
Below we look at the estimated standard deviations. The full fit and fit from total residuals estimate \(\phi_{S2S}\) well, while the values estimated using linear regression are biased low (using well recorded stations reduces the bias).
df <- data.frame(model = c('true', 'sample','full', 'dS','dS(N>=10)','dR'),
phi_s2s = c(phi_s2s_sim, sd(dS_sim),as.data.frame(VarCorr(fit_sim))$sdcor[1],
sigma(fit_sim2a), sigma(fit_sim3a), as.data.frame(VarCorr(fit_sim4))$sdcor[1]))
knitr::kable(df, digits = 5, row.names = TRUE,
caption = "Phi_S2S")| model | phi_s2s | |
|---|---|---|
| 1 | true | 0.23000 |
| 2 | sample | 0.23696 |
| 3 | full | 0.23493 |
| 4 | dS | 0.20739 |
| 5 | dS(N>=10) | 0.22793 |
| 6 | dR | 0.23493 |
2.4 Two-Step Regression
In the simulations, we consder two random effects: event terms \(\delta B\) and site terms \(\delta S\). So far, we have estimated the
random effects and their associated standard deviations in one single
step (lmer allows to fit crossed random effects, the
Bayesian methods as well). Traditionally, in GMM development often the
algorithm of Abrahamson and Youngs (1992) is used, which does not work
for crossed random effects (Similarly, package nlme does
not work for crossed random effects). In this case, one ofen first runs
a egresson wh only event terms, and then a second regression which
partitions the between-event residuals from the first regression into
\(\delta S\) and \(\delta WS\). In the following, we
investigate the differences. Again, we randomly sample even terms, site
terms, and within-event/within-site residuals, and then fit regression
models using lmer. We repeat this process several times,
and record the estimated values of coefficients and standard deviations,
as well as the confidence intervals.
tau_sim <- 0.17
phi_s2s_sim <- 0.23
phi_ss_sim <- 0.2
sds_sim <- c(phi_s2s_sim, tau_sim, phi_ss_sim)
n_sam <- 100
mat_fix <- matrix(ncol = length(coeffs), nrow = n_sam)
mat_fix2 <- matrix(ncol = length(coeffs), nrow = n_sam)
mat_ci <- matrix(nrow = n_sam, ncol = length(coeffs))
mat_ci2 <- matrix(nrow = n_sam, ncol = length(coeffs))
mat_ci_sd <- matrix(nrow = n_sam, ncol = 3)
mat_ci_sd2 <- matrix(nrow = n_sam, ncol = 3)
mat_sd <- matrix(nrow = n_sam, ncol = 3)
mat_sd2 <- matrix(nrow = n_sam, ncol = 3)
set.seed(8472)
for(i in 1:n_sam) {
dWS_sim <- rnorm(n_rec, sd = phi_ss_sim)
dS_sim <- rnorm(n_stat, sd = phi_s2s_sim)
dB_sim <- rnorm(n_eq, sd = tau_sim)
data_reg$y_sim <- as.matrix(data_reg[,names_coeffs]) %*% coeffs + dB_sim[eq] + dS_sim[stat] + dWS_sim
fit_sim <- lmer(y_sim ~ M1 + M2 + MlogR + logR + R + logVS + (1|eq) + (1|stat), data_reg)
fit_sim2 <- lmer(y_sim ~ M1 + M2 + MlogR + logR + R + logVS + (1|eq), data_reg)
ci_sim <- confint(fit_sim, level = 0.9)
ci_sim2 <- confint(fit_sim2, level = 0.9)
for(k in 1:length(coeffs)) {mat_ci[i,k] <- sum(coeffs[k] > ci_sim[k+3,1] & coeffs[k] <= ci_sim[k+3,2])}
for(k in 1:length(coeffs)) {mat_ci2[i,k] <- sum(coeffs[k] > ci_sim2[k+2,1] & coeffs[k] <= ci_sim2[k+2,2])}
for(k in 1:length(sds_sim)) {mat_ci_sd[i,k] <- sum(sds_sim[k] > ci_sim[k,1] & sds_sim[k] <= ci_sim[k,2])}
mat_fix[i,] <- fixef(fit_sim)
mat_fix2[i,] <- fixef(fit_sim2)
mat_sd[i,] <- as.data.frame(VarCorr(fit_sim))$sdcor
data_reg$dR2 <- resid(fit_sim2)
fit_sim2a <- lmer(dR2 ~ (1 | stat), data_reg)
tmp <- as.data.frame(VarCorr(fit_sim2a))$sdcor
mat_sd2[i,] <- c(tmp[1], as.data.frame(VarCorr(fit_sim2))$sdcor[1], tmp[2])
ci_sim2a <- confint(fit_sim2a, level = 0.9)
mat_ci_sd2[i,] <- c(sum(phi_s2s_sim > ci_sim2a[1,1] & phi_s2s_sim <= ci_sim2a[1,2]),
sum(tau_sim > ci_sim2[1,1] & tau_sim <= ci_sim2[1,2]),
sum(phi_ss_sim > ci_sim2a[2,1] & phi_ss_sim <= ci_sim2a[2,2]))
}Below we plot the desities of the estimated coefficients for the repeated simulations, for both the one-sep and two-step runs. Overall, on average we can recover the coefficients using both methods reasonably well, but the two-step procedure leads to somewhat wider ranges of the estimated coefficients.
df1 <- data.frame(mat_fix) %>% set_names(names_coeffs)
df1$model <- '1-step'
df2 <- data.frame(mat_fix2) %>% set_names(names_coeffs)
df2$model <- '2-step'
df <- data.frame(name = names_coeffs,
true = coeffs)
rbind(df1 %>% pivot_longer(!model),
df2 %>% pivot_longer(!model)) %>%
ggplot() +
geom_density(aes(x = value, color = model), linewidth = 1.5, key_glyph = draw_key_path) +
facet_wrap(vars(name), scales = "free") +
geom_vline(aes(xintercept = true), data = df, linewidth = 1.5) +
guides(color = guide_legend(title = NULL)) +
labs(x = '') +
theme(legend.position = c(0.8,0.2),
strip.text = element_text(size = 20))
Generally, it is important to not only o have a good estimate of how well the coefficients are estimated, but also how well we ca quantify uncertainty. Epistemc uncertainty associated with predctions is very important in PSHA. We now look how often the true coefficient lies inside the 90% confidence interval. For the one-step regression, the estimates are well calibrated (true coefficient values are inside the 90% confidence interval roughly 90% of the time), while they are not for the two-step procedure.
data.frame(rbind(colSums(mat_ci)/n_sam,
colSums(mat_ci2)/n_sam),
row.names = c('one-step','two-step')) %>%
set_names(names_coeffs) %>%
knitr::kable(digits = 5, row.names = TRUE,
caption = "Fraction how many times the estimated coefficient is inside the 90% confidence interval.")| intercept | M1 | M2 | MlogR | logR | R | logVS | |
|---|---|---|---|---|---|---|---|
| one-step | 0.91 | 0.92 | 0.88 | 0.93 | 0.91 | 0.89 | 0.86 |
| two-step | 0.71 | 0.87 | 0.90 | 0.79 | 0.66 | 0.57 | 0.28 |
Below, we plot the densities of the estimated standard deviations, with similar results and conclusions as for the coefficients.
names_sds <- c('phi_s2s','tau','phi_ss')
df1 <- data.frame(mat_sd) %>% set_names(names_sds)
df1$model <- '1-step'
df2 <- data.frame(mat_sd2) %>% set_names(names_sds)
df2$model <- '2-step'
df <- data.frame(name = names_sds,
true = sds_sim)
rbind(df1 %>% pivot_longer(!model),
df2 %>% pivot_longer(!model)) %>%
ggplot() +
geom_density(aes(x = value, color = model), linewidth = 1.5, key_glyph = draw_key_path) +
facet_wrap(vars(name), scales = "free") +
geom_vline(aes(xintercept = true), data = df, linewidth = 1.5) +
guides(color = guide_legend(title = NULL)) +
labs(x = '') +
theme(legend.position = c(0.8,0.2),
strip.text = element_text(size = 20))
For the confidence intervals of the standard deviations, again the two-step procedure s not well calibrated.
data.frame(rbind(colSums(mat_ci_sd)/n_sam,
colSums(mat_ci_sd2)/n_sam),
row.names = c('one-step','two-step')) %>%
set_names(names_sds) %>%
knitr::kable(digits = 5, row.names = TRUE,
caption = "Fraction how many times the estimated standard deviation is inside the 90% confidence interval.")| phi_s2s | tau | phi_ss | |
|---|---|---|---|
| one-step | 0.94 | 0.83 | 0.86 |
| two-step | 0.80 | 0.82 | 0.88 |
2.5 Correlations between Random Effects
Here, we briefly show that estimating correlations between random effects/residuals can be well estimated from point estimates. We simulate correlated terms from a bivariate normal distribution, perform a linear mixed-effects regression on each target variable separately, and then calculate the correlation.
The correlation coefficient is calculated as \[
\rho(X,Y) = \frac{cov(X,Y)}{\sigma_X \sigma_Y}
\] In general, in GMM modeling we are interested in the
correlations of the different terms in the model (event terms, site
terms, wthin-event/within-site residuals), which means we can only
calculate the sample covariance of the estimates of these terms. For the
standard deviations, however, we can either use the sample standard
deviations of the point estimates (which neglects uncertainty), or use
the (RE)ML estimate (which takes uncertainty in the random
effects/residuals into account). We compare calculating \(\rho\) using the standard deviations of the
point estimates in the denominator (whch iswhatis done by usig function
cor), as well as the ML estimate from lmer.
The correlations are well estimated by using point estimates, but are
underestimated when using the (RE)ML value in the denominator. The
reason is that the sample covariance of the point estimates
underestimates the true covariance.
Next to the individual correlations of the random effects and residual, we are also interested in the total correlations, which are calculated as \[\begin{equation} \rho_{IM1,IM2} = \frac{\rho_{\delta B_1, \delta B_2} \tau_1 \tau_2 + \rho_{\delta S_1, \delta S_2} \phi_{S2S,1} \phi_{S2S,2} + \rho_{\delta W_1, \delta W_2} \phi_{SS,1} \phi_{SS,2}}{\sigma_{1}\sigma_2} \label{eq: corr total} \end{equation}\] where \(IM1\) and \(IM2\) are different ground-motion intensity measures. There are again different possible choices for the use of the standard deviations. We calculate the total correlation usng the best estimate (REML), the sample standard deviation, and also as the correlation of total residuals.
tau_sim1 <- 0.4
phi_s2s_sim1 <- 0.43
phi_ss_sim1 <- 0.5
tau_sim2 <- 0.45
phi_s2s_sim2 <- 0.4
phi_ss_sim2 <- 0.55
rho_tau <- 0.95
rho_ss <- 0.9
rho_s2s <- 0.85
sigma_tot1 <- sqrt(tau_sim1^2 + phi_s2s_sim1^2 + phi_ss_sim1^2)
sigma_tot2 <- sqrt(tau_sim2^2 + phi_s2s_sim2^2 + phi_ss_sim2^2)
rho_total <- (rho_tau * tau_sim1 * tau_sim2 +
rho_s2s * phi_s2s_sim1 * phi_s2s_sim2 +
rho_ss * phi_ss_sim1 * phi_ss_sim2) /
(sigma_tot1 * sigma_tot2)
cov_tau <- matrix(c(tau_sim1^2, rho_tau * tau_sim1 * tau_sim2,
rho_tau * tau_sim1 * tau_sim2, tau_sim2^2), ncol = 2)
cov_s2s <- matrix(c(phi_s2s_sim1^2, rho_s2s * phi_s2s_sim1 * phi_s2s_sim2,
rho_s2s * phi_s2s_sim1 * phi_s2s_sim2, phi_s2s_sim2^2), ncol = 2)
cov_ss <- matrix(c(phi_ss_sim1^2, rho_ss * phi_ss_sim1 * phi_ss_sim2,
rho_ss * phi_ss_sim1 * phi_ss_sim2, phi_ss_sim2^2), ncol = 2)
eqt2 <- mvtnorm::rmvnorm(n_eq, sigma = cov_tau)
statt2 <- mvtnorm::rmvnorm(n_stat, sigma = cov_s2s)
rect2 <- mvtnorm::rmvnorm(n_rec, sigma = cov_ss)
data_reg$y_sim1 <- eqt2[eq,1] + statt2[stat,1] + rect2[,1]
data_reg$y_sim2 <- eqt2[eq,2] + statt2[stat,2] + rect2[,2]
fit_sim1 <- lmer(y_sim1 ~ (1 | eq) + (1 | stat), data_reg)
fit_sim2 <- lmer(y_sim2 ~ (1 | eq) + (1 | stat), data_reg)
dB1 <- ranef(fit_sim1)$eq$`(Intercept)`
dS1 <- ranef(fit_sim1)$stat$`(Intercept)`
dWS1 <- resid(fit_sim1)
dR1 <- data_reg$y_sim1 - predict(fit_sim1, re.form=NA)
dB2 <- ranef(fit_sim2)$eq$`(Intercept)`
dS2 <- ranef(fit_sim2)$stat$`(Intercept)`
dWS2 <- resid(fit_sim2)
dR2 <- data_reg$y_sim2 - predict(fit_sim2, re.form=NA)
sds1 <- as.data.frame(VarCorr(fit_sim1))$sdcor
sds2 <- as.data.frame(VarCorr(fit_sim2))$sdcor
sds1a <- c(sd(dS1), sd(dB1), sd(dWS1))
sds2a <- c(sd(dS2), sd(dB2), sd(dWS2))
rho_t <- (sds1[1] * sds2[1] * cor(dS1, dS2) +
sds1[2] * sds2[2] * cor(dB1, dB2) +
sds1[3] * sds2[3] * cor(dWS1, dWS2)) /
(sqrt(sum(sds1^2)) * sqrt(sum(sds2^2)))
rho_ta <- (sds1a[1] * sds2a[1] * cor(dS1, dS2) +
sds1a[2] * sds2a[2] * cor(dB1, dB2) +
sds1a[3] * sds2a[3] * cor(dWS1, dWS2)) /
(sqrt(sum(sds1a^2)) * sqrt(sum(sds2a^2)))
df <- data.frame(dS = c(rho_s2s, cor(dS1,dS2), cov(dS1,dS2)/(sd(dS1) * sd(dS2)), cov(dS1,dS2)/(sds1[1] * sds2[1])),
dB = c(rho_tau, cor(dB1,dB2), cov(dB1,dB2)/(sd(dB1) * sd(dB2)), cov(dB1,dB2)/(sds1[2] * sds2[2])),
dWS = c(rho_ss, cor(dWS1,dWS2), cov(dWS1,dWS2)/(sd(dWS1) * sd(dWS2)), cov(dWS1,dWS2)/(sds1[3] * sds2[3])),
dR = c(rho_total, cor(dR1, dR2), rho_t, rho_ta),
row.names = c('true','cor','cov/sd(point estimate)','cov()/hat()'))
knitr::kable(df, digits = 3, row.names = TRUE,
caption = "Estimated correlation coefficients.")| dS | dB | dWS | dR | |
|---|---|---|---|---|
| true | 0.850 | 0.950 | 0.900 | 0.898 |
| cor | 0.863 | 0.952 | 0.898 | 0.905 |
| cov/sd(point estimate) | 0.863 | 0.952 | 0.898 | 0.904 |
| cov()/hat() | 0.518 | 0.853 | 0.815 | 0.907 |
We can also estimate the correlations using a Bayesian model. In this case, we use a similar model as before, but pass the two target variables as data. The model estimates standard deviations, random effects, and correlations simultaneously.
data_list <- list(
N = n_rec,
NEQ = n_eq,
NSTAT = n_stat,
Y = data_reg[,c('y_sim1','y_sim2')],
eq = eq,
stat = stat
)
mod <- cmdstan_model(file.path('./Git/MixedModels_Biases/', 'stan', 'gmm_partition_corrre_cond.stan'))
fit_stan <- mod$sample(
data = data_list,
seed = 8472,
chains = 4,
iter_sampling = 200,
iter_warmup = 300,
refresh = 100,
max_treedepth = 10,
adapt_delta = 0.8,
parallel_chains = 2,
show_exceptions = FALSE
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##
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## Mean chain execution time: 644.8 seconds.
## Total execution time: 1297.3 seconds.## Processing csv files: /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpBTxlOU/gmm_partition_corrre_cond-202406170943-1-115853.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpBTxlOU/gmm_partition_corrre_cond-202406170943-2-115853.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpBTxlOU/gmm_partition_corrre_cond-202406170943-3-115853.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpBTxlOU/gmm_partition_corrre_cond-202406170943-4-115853.csv
##
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## Treedepth satisfactory for all transitions.
##
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## No divergent transitions found.
##
## Checking E-BFMI - sampler transitions HMC potential energy.
## E-BFMI satisfactory.
##
## Effective sample size satisfactory.
##
## Split R-hat values satisfactory all parameters.
##
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## $status
## [1] 0
##
## $stdout
## [1] "Processing csv files: /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpBTxlOU/gmm_partition_corrre_cond-202406170943-1-115853.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpBTxlOU/gmm_partition_corrre_cond-202406170943-2-115853.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpBTxlOU/gmm_partition_corrre_cond-202406170943-3-115853.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpBTxlOU/gmm_partition_corrre_cond-202406170943-4-115853.csv\n\nChecking sampler transitions treedepth.\nTreedepth satisfactory for all transitions.\n\nChecking sampler transitions for divergences.\nNo divergent transitions found.\n\nChecking E-BFMI - sampler transitions HMC potential energy.\nE-BFMI satisfactory.\n\nEffective sample size satisfactory.\n\nSplit R-hat values satisfactory all parameters.\n\nProcessing complete, no problems detected.\n"
##
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## [1] ""
##
## $timeout
## [1] FALSE## $num_divergent
## [1] 0 0 0 0
##
## $num_max_treedepth
## [1] 0 0 0 0
##
## $ebfmi
## [1] 0.7973268 0.8146631 1.0102995 1.0385106For a real analsswe should probably run the chains with more observations, but for a quick demonstration this is enough. First we look ata summary of estimated standard deviations and correlation coefficients. Overall, the fit is ok.
## # A tibble: 9 × 10
## variable mean median sd mad q5 q95 rhat ess_bulk ess_tail
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 tau[1] 0.415 0.414 0.0198 0.0202 0.383 0.448 1.00 781. 682.
## 2 tau[2] 0.472 0.472 0.0227 0.0232 0.437 0.509 1.01 765. 683.
## 3 phi_ss[1] 0.500 0.500 0.00340 0.00347 0.495 0.506 1.00 1055. 699.
## 4 phi_ss[2] 0.551 0.551 0.00376 0.00380 0.545 0.557 1.00 1160. 666.
## 5 phi_s2s[1] 0.417 0.417 0.0110 0.0109 0.399 0.435 1.01 414. 629.
## 6 phi_s2s[2] 0.397 0.397 0.0114 0.0113 0.377 0.416 1.02 324. 452.
## 7 rho_rec 0.900 0.900 0.00189 0.00192 0.897 0.903 0.999 1477. 638.
## 8 rho_eq 0.955 0.956 0.00611 0.00615 0.945 0.965 1.01 908. 726.
## 9 rho_stat 0.846 0.846 0.0120 0.0121 0.826 0.865 1.01 455. 531.Below we look at the posterior distributions of the estimated
correlations. The vertical black line is the true value, and the solid
red line is the value estimated from the randomeffects ore residuals.
The dashed lines are the 90% confidence limits, estimated using Fisher’s
z-transformation (Fisher 1915). We also show the
posterior distribution of the total correlation, which can be calculated
from the samples for the individual correlations and standard deviatons
(we use the rvar data type from the posterior package,
which allos to easily perform calculations on samples from the
posterior).
In general, the fit and associated uncertainty looks good, for both methods.
inverse_ztransform <- function(rho) {
return((exp(2 * rho) - 1) / (exp(2 * rho) + 1))
}
ztransform <- function(rho) {
0.5 * log((1+rho) / (1-rho))
}
calc_ci <- function(rho, n, level = 0.9) {
scale <- abs(qnorm((1 - level)/2))
z <- ztransform(rho)
se <- 1/sqrt(n - 3)
return(c(inverse_ztransform(z - scale * se), inverse_ztransform(z + scale * se)))
}
rho_total_stan <- (rv$phi_ss[1] * rv$phi_ss[2] * rv$rho_rec +
rv$phi_s2s[1] * rv$phi_s2s[2] * rv$rho_stat +
rv$tau[1] * rv$tau[2] * rv$rho_eq) /
(sqrt(rv$phi_ss[1]^2 + rv$phi_s2s[1]^2 + rv$tau[1]^2) * sqrt(rv$phi_ss[2]^2 + rv$phi_s2s[2]^2 + rv$tau[2]^2))
patchwork::wrap_plots(
mcmc_dens(draws, pars = 'rho_eq') +
vline_at(rho_tau, linewidth = 1.5) +
vline_at(cor(dB1,dB2), linewidth = 1.5, color = 'red') +
vline_at(calc_ci(cor(dB1,dB2), n_eq), linewidth = 1.5, color = 'red', linetype = 'dashed'),
mcmc_dens(draws, pars = 'rho_stat') +
vline_at(rho_s2s, linewidth = 1.5) +
vline_at(cor(dS1,dS2), linewidth = 1.5, color = 'red') +
vline_at(calc_ci(cor(dS1,dS2), n_stat), linewidth = 1.5, color = 'red', linetype = 'dashed'),
mcmc_dens(draws, pars = 'rho_rec') +
vline_at(rho_ss, linewidth = 1.5) +
vline_at(cor(dWS1,dWS2), linewidth = 1.5, color = 'red') +
vline_at(calc_ci(cor(dWS1,dWS2), n_rec), linewidth = 1.5, color = 'red', linetype = 'dashed'),
mcmc_dens(as_draws(rho_total_stan)) +
vline_at(rho_total, linewidth = 1.5) +
vline_at(rho_t, linewidth = 1.5, color = 'red') +
labs(x = 'rho_total')
)
In the plot above, the posterior distribution of the total correlation coefficient is calclatedform the posterior samples of the standard deviations and correlatio coefficients, and thus also accounts for uncertainty of the standard devations. We can compare this with a calculation that uses a point estimate for the standard deviations. In this case, the uncertainty of the total correlation is slightly lower, but the difference is small and shouldn’t matter in practice.
rho_total_stan2 <- (mean(rv$phi_ss[1]) * mean(rv$phi_ss[2]) * rv$rho_rec +
mean(rv$phi_s2s[1]) * mean(rv$phi_s2s[2]) * rv$rho_stat +
mean(rv$tau[1]) * mean(rv$tau[2]) * rv$rho_eq) /
(sqrt(mean(rv$phi_ss[1])^2 + mean(rv$phi_s2s[1])^2 + mean(rv$tau[1])^2) *
sqrt(mean(rv$phi_ss[2])^2 + mean(rv$phi_s2s[2])^2 + mean(rv$tau[2])^2))
c(rho_total_stan, rho_total_stan2)## rvar<200,4>[2] mean ± sd:
## [1] 0.9 ± 0.0041 0.9 ± 0.00362.6 Correlation with e.g. Stress Drop
Often, correlations between event terms (typically for PGA) and stress drops are calcualted and investigated. Below, we investigate such a case, by simulating event terms and stress drop from a bivariate normal distribution. We then fit a mixed-effects model, and estimate the correlations between the estimated event terms and stress drops.
tau_sim <- 0.4
phi_s2s_sim <- 0.43
phi_ss_sim <- 0.5
rho <- 0.7
tau2 <- 1
cov_tau <- matrix(c(tau_sim^2, rho * tau_sim * tau2,
rho * tau_sim * tau2, tau2^2), ncol = 2)
set.seed(5618)
deltaWS_sim <- rnorm(n_rec, sd = phi_ss_sim)
deltaS_sim <- rnorm(n_stat, sd = phi_s2s_sim)
eqt2 <- mvtnorm::rmvnorm(n_eq, sigma = cov_tau)
y_sim <- eqt2[eq,1] + deltaS_sim[stat] + deltaWS_sim
data_reg$y_sim <- y_sim
fit_lmer_sim <- lmer(y_sim ~ (1 | eq) + (1 | stat), data_reg)
deltaB_lmer <- ranef(fit_lmer_sim)$eq$`(Intercept)`
cor(deltaB_lmer, eqt2[,2])## [1] 0.6591753The fit by itself is ok, but in this case the correlation is underestimated. For the paper, we have repeated this simulation 100 times, and on average we observe a bias.
In the following, we fit a Stan model to the data in which we directly estimate the correlation between the event terns and the other event related variable \(E\)during te model fitting phase, i.e. during partitioning. This implicitly takes the uncertainty in the estimated event terms into account. We model the two variables as a bivariate normal distribution, and use the conditional distribution of \(\delta B\) given \(E\) as the prior distribution for \(\delta B\) \[ \begin{aligned} \delta B &\sim N(\mu_s, \tau_s) \\ \mu_s &= \frac{\tau}{\sigma_{E}} \; \rho \;E \\ \tau_s &= \sqrt{(1 - \rho^2) \tau^2} \end{aligned} \]
mod <- cmdstan_model(file.path('./Git/MixedModels_Biases/', 'stan', 'gmm_partition_wvar_corr.stan'))
data_list_cor <- list(
N = n_rec,
NEQ = n_eq,
NSTAT = n_stat,
Y = data_reg$y_sim,
E = eqt2[,2],
eq = eq,
stat = stat,
alpha = c(1,1,1)
)
fit <- mod$sample(
data = data_list_cor,
seed = 8472,
chains = 4,
iter_sampling = 200,
iter_warmup = 200,
refresh = 100,
max_treedepth = 10,
adapt_delta = 0.8,
parallel_chains = 2,
show_exceptions = FALSE
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## Total execution time: 184.2 seconds.## Processing csv files: /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpTUiZ6p/gmm_partition_wvar_corr-202406171507-1-80c7c2.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpTUiZ6p/gmm_partition_wvar_corr-202406171507-2-80c7c2.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpTUiZ6p/gmm_partition_wvar_corr-202406171507-3-80c7c2.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpTUiZ6p/gmm_partition_wvar_corr-202406171507-4-80c7c2.csv
##
## Checking sampler transitions treedepth.
## Treedepth satisfactory for all transitions.
##
## Checking sampler transitions for divergences.
## No divergent transitions found.
##
## Checking E-BFMI - sampler transitions HMC potential energy.
## E-BFMI satisfactory.
##
## Effective sample size satisfactory.
##
## Split R-hat values satisfactory all parameters.
##
## Processing complete, no problems detected.
## $status
## [1] 0
##
## $stdout
## [1] "Processing csv files: /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpTUiZ6p/gmm_partition_wvar_corr-202406171507-1-80c7c2.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpTUiZ6p/gmm_partition_wvar_corr-202406171507-2-80c7c2.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpTUiZ6p/gmm_partition_wvar_corr-202406171507-3-80c7c2.csv, /var/folders/p3/r7vrsk6n2d15709vgcky_y880000gn/T/RtmpTUiZ6p/gmm_partition_wvar_corr-202406171507-4-80c7c2.csv\n\nChecking sampler transitions treedepth.\nTreedepth satisfactory for all transitions.\n\nChecking sampler transitions for divergences.\nNo divergent transitions found.\n\nChecking E-BFMI - sampler transitions HMC potential energy.\nE-BFMI satisfactory.\n\nEffective sample size satisfactory.\n\nSplit R-hat values satisfactory all parameters.\n\nProcessing complete, no problems detected.\n"
##
## $stderr
## [1] ""
##
## $timeout
## [1] FALSE## $num_divergent
## [1] 0 0 0 0
##
## $num_max_treedepth
## [1] 0 0 0 0
##
## $ebfmi
## [1] 0.8160571 0.8943540 0.7220071 0.8905378draws_corr <- fit$draws()
summarise_draws(subset(draws_corr, variable = c('rho', 'phi', 'tau'), regex = TRUE))## # A tibble: 5 × 10
## variable mean median sd mad q5 q95 rhat ess_bulk ess_tail
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 rho 0.686 0.687 0.0325 0.0323 0.629 0.736 1.00 872. 717.
## 2 phi_ss 0.497 0.497 0.00335 0.00331 0.492 0.503 1.01 649. 528.
## 3 phi_s2s 0.441 0.441 0.0115 0.0119 0.422 0.459 1.00 568. 684.
## 4 tau2 0.992 0.991 0.0434 0.0453 0.926 1.07 1.00 867. 657.
## 5 tau 0.422 0.420 0.0205 0.0207 0.389 0.456 1.00 590. 466.The fit looks good, and we also get a good estimate of \(\rho\).
Below, we show the posterior distribution of the correlation
coefficient \(\rho\), together with the
true value (black), the correlation between point estimates \(\widehat{\delta B}\) and \(E\) from lmer (red), and the
correlation between the mean event terms (point estimates) from the Stan
fit (blue). The black dashed line is the mean of he posterior
distribution of rho.
mcmc_hist(draws_corr, pars = 'rho') +
vline_at(rho, linewidth = 1.5) +
vline_at(colMeans(subset(as_draws_matrix(draws_corr), variable = 'rho', regex = FALSE)),
linewidth = 1.5, linetype = 'dashed') +
vline_at(cor(deltaB_lmer, eqt2[,2]), linewidth = 1.5, color = 'red') +
vline_at(cor(eqt2[,2], colMeans(subset(as_draws_matrix(draws_corr), variable = 'eqterm', regex = TRUE))),
linewidth = 1.5, color = 'blue')## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
3 Simulations using Italian Data
Now, we use the Italian data of Lanzano et al. (2019) (see also Caramenti et al. (2022)) for simulations with spatial/nonergodic models. This data was used in N. Kuehn (2023) for nonergodic model comparison, which made it easy to set up.
First, we read and prepare the data.
data_it <- read.csv(file.path('./Git/MixedModels_Biases/','/data','italian_data_pga_id_utm_stat.csv'))
# Set linear predictors
mh = 5.5
mref = 5.324
h = 6.924
attach(data_it)
b1 = (mag-mh)*(mag<=mh)
b2 = (mag-mh)*(mag>mh)
c1 = (mag-mref)*log10(sqrt(JB_complete^2+h^2))
c2 = log10(sqrt(JB_complete^2+h^2))
c3 = sqrt(JB_complete^2+h^2)
f1 = as.numeric(fm_type_code == "SS")
f2 = as.numeric(fm_type_code == "TF")
k = log10(vs30/800)*(vs30<=1500)+log10(1500/800)*(vs30>1500)
y = log10(rotD50_pga)
detach(data_it)
n_rec <- length(b1)
eq <- data_it$EQID
stat <- data_it$STATID
n_eq <- max(eq)
n_stat <- max(stat)
n_rec <- nrow(data_it)
mageq <- unique(data_it[,c('EQID','mag')])[,2]
data_reg <- data.frame(Y = y,
M1 = b1,
M2 = b2,
MlogR = c1,
logR = c2,
R = c3,
Fss = f1,
Frv = f2,
logVS = k,
eq = eq,
stat = stat,
intercept = 1,
M = data_it$mag
)
# coefficients from ITA18
coeffs <- c(3.421046409, 0.193954090, -0.021982777, 0.287149291,
-1.405635476, -0.002911264, 0.085983743, 0.010500239,
-0.394575970)
names_coeffs <- c("intercept", "M1", "M2", "MlogR", "logR", "R", "Fss", "Frv", "logVS")p1 <- ggplot(data_it) +
geom_point(aes(x = JB_complete, y = mag)) +
scale_x_log10(breaks = breaks, minor_breaks = minor_breaks)
p2 <- ggplot(unique(data_it[,c('EQID','mag')])) +
geom_histogram(aes(x = mag))
patchwork::wrap_plots(p1, p2)## Warning in scale_x_log10(breaks = breaks, minor_breaks = minor_breaks): log-10
## transformation introduced infinite values.## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
We
simulate data for some spatial models on the Italian data, and we use
INLA (https://www.r-inla.org/) to estimate the models. Below,
we set the penalized complexity prior (Simpson et al. 2017) for the standard
deviations, used throughout.
prior_prec_tau <- list(prec = list(prior = 'pc.prec', param = c(0.3, 0.01)))
prior_prec_phiS2S <- list(prec = list(prior = 'pc.prec', param = c(0.3, 0.01)))
prior_prec_phiSS <- list(prec = list(prior = 'pc.prec', param = c(0.3, 0.01)))3.1 Spatial correlations of Site Terms
In nonergodic models, site terms are often modeled as spatially correlated. The spatial correlation structure can be assessed from point estimates of the site terms. Here, we simulate some data with spatially correlated site terms, to check whether we can get the model parameters back.
We use the Mat'ern covariance function for the spatial correlation of the site terms, which is defined below. In general, we follow Krainski et al. (2019) when seting up the spatial models.
cMatern <- function(h, nu, kappa) {
ifelse(h > 0, besselK(h * kappa, nu) * (h * kappa)^nu /
(gamma(nu) * 2^(nu - 1)), 1)
}
# Function to sample from zero mean multivariate normal
rmvnorm0 <- function(n, cov, R = NULL) {
if (is.null(R))
R <- chol(cov)
return(crossprod(R, matrix(rnorm(n * ncol(R)), ncol(R))))
}Next we define the coefficients, spatial range, and standard deviations for the simulation.
# unique station coordinates
co_stat_utm <- unique(data_it[,c("STATID", "X_stat","Y_stat")])[,c(2,3)]
range <- 30 # spatial range
nu <- 1
kappa <- sqrt(8*nu)/range
# standard deviations
wvar <- 0.65 # variance ratio for phi_s2s
tau_sim <- 0.17
phi_s2s_sim <- 0.23
phi_s2s_0 <- sqrt((1 - wvar) * phi_s2s_sim^2)
phi_s2s_c <- sqrt(wvar * phi_s2s_sim^2)
phi_ss_sim <- 0.2
cov <- phi_s2s_c^2 * cMatern(as.matrix(dist(co_stat_utm)), nu, kappa) + diag(10^-9, n_stat)To use R-INLA with the ``stochastic partial differential equation’’ (SPDE) approach, we need to define a mesh.
max.edge2 <- 5
bound.outer2 <- 40
mesh = inla.mesh.2d(loc=co_stat_utm,
max.edge = c(1,5)*max.edge2,
cutoff = max.edge2,
offset = c(5 * max.edge2, bound.outer2))
print(mesh$n) # print number of mesh nodes## [1] 26702Now we define priors for the standard deviations (based on Simpson et al. (2017)), the SPDE prior, the projecion
matrix A, and the formula.
spde_stat <- inla.spde2.pcmatern(
# Mesh and smoothness parameter
mesh = mesh, alpha = 2,
# P(practic.range < 0.3) = 0.5
prior.range = c(100, 0.9),
# P(sigma > 1) = 0.01
prior.sigma = c(.3, 0.01))
A_stat <- inla.spde.make.A(mesh, loc = as.matrix(co_stat_utm[stat,]))
A_stat_unique <- inla.spde.make.A(mesh, loc = as.matrix(co_stat_utm))
idx_stat <- inla.spde.make.index("idx_stat",spde_stat$n.spde)
# formula to be used on the total residuals
form_spatial_total <- y ~ 0 + intercept +
f(eq, model = "iid", hyper = prior_prec_tau) +
f(stat, model = "iid",hyper = prior_prec_phiS2S) +
f(idx_stat, model = spde_stat)
# formula for the full fit
form_spatial_stat <- y ~ 0 + intercept +
M1 + M2 + MlogR + logR + R + Fss + Frv + logVS +
f(eq, model = "iid", hyper = prior_prec_tau) +
f(stat, model = "iid",hyper = prior_prec_phiS2S) +
f(idx_stat, model = spde_stat)
# formula for fit from site terms
form_spatial_stat_u <- y ~ 0 + intercept + f(idx_stat, model = spde_stat)Now, we sample the event terms, site terms, spatially correlated site
terms, and within-event/within-site residuals, and combine them with the
median predictions from ITA18. We then fit a lmer model to
the data.
set.seed(8472)
dB_sim <- rnorm(n_eq, mean =0, sd = tau_sim)
dS_sim <- rnorm(n_stat, mean =0, sd = phi_s2s_0)
dWS_sim <- rnorm(n_rec, mean = 0, sd = phi_ss_sim)
data_reg$y_sim <- as.matrix(data_reg[,names_coeffs]) %*% coeffs +
dB_sim[eq] + dS_sim[stat] + dWS_sim + rmvnorm0(1, cov)[stat]
fit_sim <- lmer(y_sim ~ M1 + M2 + MlogR + logR + R + Fss + Frv + logVS + (1|eq) + (1|stat), data_reg)
dR_lmer <- data_reg$y_sim - predict(fit_sim, re.form=NA)
dS_lmer <- ranef(fit_sim)$stat$`(Intercept)`Now, we fit the Inla models. We fit the full model (fixed and random
effects), a model on the total residuals, and a model on the site terms
from the lmer fit.
#### full model
# create the stack
stk_spatial_stat <- inla.stack(
data = list(y = data_reg$y_sim),
A = list(A_stat, 1),
effects = list(idx_stat = idx_stat,
data_reg
))
fit_inla_spatial_stat <- inla(form_spatial_stat,
data = inla.stack.data(stk_spatial_stat),
control.predictor = list(A = inla.stack.A(stk_spatial_stat)),
family="gaussian",
control.family = list(hyper = prior_prec_phiSS),
control.inla = list(int.strategy = "eb", strategy = "gaussian"),
quantiles = c(0.05, 0.5, 0.95)
)
### ft from site terms
data_reg_stat <- data.frame(
dS = dS_lmer,
intercept = 1
)
# create the stack
stk_spatial_stat_u <- inla.stack(
data = list(y = data_reg_stat$dS),
A = list(A_stat_unique, 1),
effects = list(idx_stat = idx_stat,
data_reg_stat
))
fit_inla_spatial_stat_u <- inla(form_spatial_stat_u,
data = inla.stack.data(stk_spatial_stat_u),
control.predictor = list(A = inla.stack.A(stk_spatial_stat_u)),
family="gaussian",
control.family = list(hyper = prior_prec_phiS2S),
control.inla = list(int.strategy = "eb", strategy = "gaussian"),
quantiles = c(0.05, 0.5, 0.95)
)
#### fit from total residuals
data_reg$deltaR <- dR_lmer
# create the stack
stk_spatial_total <- inla.stack(
data = list(y = data_reg$deltaR),
A = list(A_stat, 1),
effects = list(idx_stat = idx_stat,
data_reg
))
fit_inla_spatial_total <- inla(form_spatial_total,
data = inla.stack.data(stk_spatial_total),
control.predictor = list(A = inla.stack.A(stk_spatial_total)),
family="gaussian",
control.family = list(hyper = prior_prec_phiSS),
control.inla = list(int.strategy = "eb", strategy = "gaussian"),
quantiles = c(0.05, 0.5, 0.95)
)Below, we plot the posterior distributions of the spatial range as well as the associated standard deviation.
p1 <- rbind(
data.frame(inla.tmarginal(function(x) exp(x),
fit_inla_spatial_stat$internal.marginals.hyperpar$`log(Range) for idx_stat`),
mod = "full"),
data.frame(inla.tmarginal(function(x) exp(x),
fit_inla_spatial_stat_u$internal.marginals.hyperpar$`log(Range) for idx_stat`),
mod = "dS"),
data.frame(inla.tmarginal(function(x) exp(x),
fit_inla_spatial_total$internal.marginals.hyperpar$`log(Range) for idx_stat`),
mod = "dR")
) %>%
ggplot() +
geom_line(aes(x = x, y = y, color = mod), linewidth = 1.5) +
geom_vline(xintercept = range, linewidth = 1.5) +
labs(x = 'spatial range (km)', 'density') +
theme(legend.position = c(0.8,0.8)) +
guides(color = guide_legend(title = NULL))
p2 <- rbind(
data.frame(inla.tmarginal(function(x) exp(x),
fit_inla_spatial_stat$internal.marginals.hyperpar$`log(Stdev) for idx_stat`),
mod = "full", type = 'phi_s2s_c'),
data.frame(inla.tmarginal(function(x) exp(x),
fit_inla_spatial_stat_u$internal.marginals.hyperpar$`log(Stdev) for idx_stat`),
mod = "dS", type = 'phi_s2s_c'),
data.frame(inla.tmarginal(function(x) exp(x),
fit_inla_spatial_total$internal.marginals.hyperpar$`log(Stdev) for idx_stat`),
mod = "dR", type = 'phi_s2s_c'),
data.frame(inla.tmarginal(function(x) sqrt(exp(-x)),
fit_inla_spatial_stat$internal.marginals.hyperpar$`Log precision for stat`),
mod = "full", type = 'phi_s2s_0'),
data.frame(inla.tmarginal(function(x) sqrt(exp(-x)),
fit_inla_spatial_stat_u$internal.marginals.hyperpar$`Log precision for the Gaussian observations`),
mod = "dS", type = 'phi_s2s_0'),
data.frame(inla.tmarginal(function(x) sqrt(exp(-x)),
fit_inla_spatial_total$internal.marginals.hyperpar$`Log precision for stat`),
mod = "dR", type = 'phi_s2s_0')
) %>%
ggplot() +
geom_line(aes(x = x, y = y, color = mod, linetype = type), linewidth = 1.5) +
geom_vline(xintercept = phi_s2s_c, linewidth = 1.5, linetype = 'dashed') +
geom_vline(xintercept = phi_s2s_0, linewidth = 1.5) +
labs(x = 'phi_S2S_c', 'density') +
theme(legend.position = c(0.8,0.8)) +
guides(color = guide_legend(title = NULL), linetype = guide_legend(title = NULL))
patchwork::wrap_plots(p1, p2)
We can see that the spatial range is quite well estimated for all approaches, and that the full model and the model based on total residuals give almost the same results. The model based on site terms does not lead to good results for the standard deviations, in particular for \(\phi_{S2S,c}\), which is severely underestimated. The relative sizes of standard devations based on the fit from \(\delta S\) are wrongly estimated.
Below, we plot differences predictions of the spatially correlated site terms between the different models. We show differences with respect to the full model. As we can see, there are strong differences in the predicted means of the spatially correlated site terms of the full model and the model estmated from site terms.
diff <- fit_inla_spatial_stat_u$summary.random$idx_stat$mean - fit_inla_spatial_stat$summary.random$idx_stat$mean
diff2 <- fit_inla_spatial_total$summary.random$idx_stat$mean - fit_inla_spatial_stat$summary.random$idx_stat$mean
p1 <- ggplot() + inlabru::gg(mesh, color = diff, nx = 500, ny = 500) +
labs(x="X (km)", y="Y (km)", title = "Difference in Mean Predictions",
subtitle = "dS - full") +
scale_fill_gradient2(name = "", limits = c(-0.15,0.15))
p2 <- ggplot() + inlabru::gg(mesh, color = diff2, nx = 500, ny = 500) +
labs(x="X (km)", y="Y (km)", title = "Difference in Mean Predictions",
subtitle = "dR - full") +
scale_fill_gradient2(name = "", limits = c(-0.15,0.15))
patchwork::wrap_plots(p1, p2)
3.2 Cell-specific attenuation
In this section, we simulate data based on the cell-specific attenuation model Dawood and Rodriguez-Marek (2013), and estimate the model parameters using the total model, as well as from within-event/within-site residuals.
First, we read in the cell-specific distances, and some definitions.
# read in cell-specific attenuation
data_dm <- rstan::read_rdump(file.path('./Git/MixedModels_Biases/','/data','dm_25x25.Rdata'))
dm_sparse <- as(data_dm$RC,"dgCMatrix") / 100 # divide by 100 to avoid small values
n_cell <- data_dm$NCELL
prior_prec_cell <- list(prec = list(prior = 'pc.prec', param = c(0.5, 0.01)))
data_reg$idx_cell <- 1:n_recNow we define the parameters for the simulation, and sample.
tau_sim <- 0.17
phi_s2s_sim <- 0.2
phi_ss_sim <- 0.18
sigma_cell_sim <- 0.35
set.seed(5618)
dB_sim <- rnorm(n_eq, mean =0, sd = tau_sim)
dS_sim <- rnorm(n_stat, mean =0, sd =phi_s2s_sim)
dWS_sim <- rnorm(n_rec, mean = 0, sd = phi_ss_sim)
dC_sim <- rnorm(n_cell, mean = 0, sd = sigma_cell_sim)
data_reg$y_sim <- as.numeric(as.matrix(data_reg[,names_coeffs]) %*% coeffs + dm_sparse %*% dC_sim +
dB_sim[eq] + dS_sim[stat] + dWS_sim)Now we fit the different models using Inla.
# full fit
fit_sim_cell <- inla(y_sim ~ 0 + intercept + M1 + M2 + logR + MlogR + R + Fss + Frv + logVS +
f(eq, model = "iid", hyper = prior_prec_tau) +
f(stat, model = "iid",hyper = prior_prec_phiS2S) +
f(idx_cell, model = "z", Z = dm_sparse, hyper = prior_prec_cell),
data = data_reg,
family="gaussian",
control.family = list(hyper = prior_prec_phiSS),
quantiles = c(0.05,0.5,0.95)
)## Warning in inla.model.properties.generic(inla.trim.family(model), mm[names(mm) == : Model 'z' in section 'latent' is marked as 'experimental'; changes may appear at any time.
## Use this model with extra care!!! Further warnings are disabled.fit_sim <- lmer(y_sim ~ M1 + M2 + logR + MlogR + R + Fss + Frv + logVS + (1|eq) + (1|stat), data_reg)
data_reg$dWS_lmer <- data_reg$y_sim - predict(fit_sim)
fit_sim_cell_dws <- inla(dWS_lmer ~ 0 + intercept +
f(idx_cell, model = "z", Z = dm_sparse, hyper = prior_prec_cell),
data = data_reg,
family="gaussian",
control.family = list(hyper = prior_prec_phiSS),
quantiles = c(0.05,0.5,0.95)
)
# total residuals, but also take out linear R term
data_reg$dR_lmer <- data_reg$y_sim - (predict(fit_sim,re.form=NA) -
fixef(fit_sim)[6] * data_reg$R)
fit_sim_cell_dR <- inla(dR_lmer ~ 0 + intercept + R +
f(eq, model = "iid", hyper = prior_prec_tau) +
f(stat, model = "iid",hyper = prior_prec_phiS2S) +
f(idx_cell, model = "z", Z = dm_sparse, hyper = prior_prec_cell),
data = data_reg,
family="gaussian",
control.family = list(hyper = prior_prec_phiSS),
quantiles = c(0.05,0.5,0.95)
)Ths shows the posterior distribution of the standard deviation of the cell-specific attenuation coefficients, which is understimated from \(\delta WS\).
rbind(data.frame(inla.tmarginal(function(x) sqrt(exp(-x)),
fit_sim_cell$internal.marginals.hyperpar$`Log precision for idx_cell`),
mod = "full"),
data.frame(inla.tmarginal(function(x) sqrt(exp(-x)),
fit_sim_cell_dws$internal.marginals.hyperpar$`Log precision for idx_cell`),
mod = "dWS"),
data.frame(inla.tmarginal(function(x) sqrt(exp(-x)),
fit_sim_cell_dR$internal.marginals.hyperpar$`Log precision for idx_cell`),
mod = "dR")
) %>%
ggplot() +
geom_line(aes(x = x, y = y, color = mod), linewidth = 1.5) +
geom_vline(xintercept = sigma_cell_sim, linewidth = 1.5) +
labs(x = 'sigma_cell', 'density') +
theme(legend.position = c(0.8,0.8)) +
guides(color = guide_legend(title = NULL))
4 Plots of Repeated Simulations
In the paper, we show results from many repeated simulations, typically as density plots of estimated parameters. The simulations are carried out using code as in this document, looping over the simulations and keeping the estimated parameters. Here, we generate some of the plots of the paper. Code to run the simulations can be found at https://github.com/nikuehn/MixedModels_Biases/tree/main/r.
4.1 Simulations with Homoscedastic Standard Deviations
# Results for simulations based on CB14 data
load(file = file.path('./Git/MixedModels_Biases/', 'results', 'results_sim1_CB.Rdata'))
# simulation
tau_sim <- 0.4
phi_s2s_sim <- 0.43
phi_ss_sim <- 0.5
# look at values of standard deviations estimated from lmer and Stan (using mean and median of posterior)
df <- data.frame(res_val) |>
set_names(c('phis2s_lmer_mode','tau_lmer_mode','phiss_lmer_mode',
'phis2s_stan_mean','tau_stan_mean','phiss_stan_mean',
'phis2s_stan_median','tau_stan_median','phiss_stan_median'))
knitr::kable(head(df), digits = 5, row.names = TRUE,
caption = "Comparison of standard deviation estimates.")| phis2s_lmer_mode | tau_lmer_mode | phiss_lmer_mode | phis2s_stan_mean | tau_stan_mean | phiss_stan_mean | phis2s_stan_median | tau_stan_median | phiss_stan_median | |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 0.44047 | 0.41113 | 0.49731 | 0.44057 | 0.41290 | 0.49736 | 0.44043 | 0.41206 | 0.49739 |
| 2 | 0.43806 | 0.38546 | 0.50110 | 0.43856 | 0.38625 | 0.50106 | 0.43880 | 0.38556 | 0.50103 |
| 3 | 0.41809 | 0.39821 | 0.49917 | 0.41878 | 0.40063 | 0.49908 | 0.41830 | 0.40080 | 0.49911 |
| 4 | 0.44819 | 0.39679 | 0.49623 | 0.44831 | 0.39811 | 0.49633 | 0.44800 | 0.39716 | 0.49623 |
| 5 | 0.42661 | 0.37745 | 0.49996 | 0.42682 | 0.37868 | 0.49990 | 0.42693 | 0.37818 | 0.49985 |
| 6 | 0.43857 | 0.42567 | 0.50033 | 0.43869 | 0.42727 | 0.50032 | 0.43828 | 0.42638 | 0.50028 |
p1 <- data.frame(res_val[,c(1,4,7)], res_sd[,c(1,4)]) %>%
set_names(c('lmer_max', 'stan_mean','stan_median','lmer_sd(dS)','stan_sd(dS)')) %>%
pivot_longer(everything(), names_sep = '_', names_to = c('model','type')) %>%
ggplot() +
geom_density(aes(x = value, color = model, linetype = type), linewidth = 1.5, key_glyph = draw_key_path) +
labs(x = expression(paste(widehat(phi)[S2S]))) +
guides(color = guide_legend(title = NULL), linetype = guide_legend(title = NULL)) +
theme(legend.position = 'none',
legend.key.width = unit(2,'cm')) +
geom_vline(xintercept = phi_s2s_sim, linewidth = 1.5) +
scale_linetype_manual(values = c(1,2,3,4),
labels = c('max','mean','median',TeX("sd($\\delta S$)"))) +
scale_color_manual(values = c('red','blue'))
p2 <- data.frame(res_val[,c(2,5,8)], res_sd[,c(2,5)]) %>%
set_names(c('lmer_max', 'stan_mean','stan_median','lmer_sd(dB)','stan_sd(dB)')) %>%
pivot_longer(everything(), names_sep = '_', names_to = c('model','type')) %>%
ggplot() +
geom_density(aes(x = value, color = model, linetype = type), linewidth = 1.5, key_glyph = draw_key_path) +
labs(x = expression(widehat(tau))) +
guides(color = guide_legend(title = NULL), linetype = guide_legend(title = NULL)) +
theme(legend.position = c(0.3,0.65),
legend.key.width = unit(1.5,'cm')) +
geom_vline(xintercept = tau_sim, linewidth = 1.5) +
scale_linetype_manual(values = c(1,2,3,4),
labels = c('max','mean','median',TeX("sd($\\delta B$)"))) +
scale_color_manual(values = c('red','blue'))
p3 <- data.frame(res_val[,c(3,6,9)], res_sd[,c(3,6)]) %>%
set_names(c('lmer_max', 'stan_mean','stan_median','lmer_sd(dWS)','stan_sd(dWS)')) %>%
pivot_longer(everything(), names_sep = '_', names_to = c('model','type')) %>%
ggplot() +
geom_density(aes(x = value, color = model, linetype = type), linewidth = 1.5, key_glyph = draw_key_path) +
labs(x = expression(paste(widehat(phi)[SS]))) +
guides(color = guide_legend(title = NULL), linetype = guide_legend(title = NULL)) +
theme(legend.position = 'none',
legend.key.width = unit(2,'cm')) +
geom_vline(xintercept = phi_ss_sim, linewidth = 1.5) +
scale_linetype_manual(values = c(1,2,3,4),
labels = c('max','mean','median',TeX("sd($\\delta WS$)"))) +
scale_color_manual(values = c('red','blue'))
leg <- ggpubr::get_legend(p2)
patchwork::wrap_plots(p1,p2 + theme(legend.position = 'none'),p3,ggpubr::as_ggplot(leg), ncol = 2)
4.2 Heteroscedastic Standard Deviations
# Results for simulations based on CB14 data
load(file = file.path('./Git/MixedModels_Biases/', 'results', 'results_sim2_heteroscedastic_coeff_CB.Rdata'))
load(file = file.path('./Git/MixedModels_Biases/', 'results', 'results_sim2_heteroscedastic_coeff_stan2_CB.Rdata'))
coeffs <- c(3.421046409, 0.193954090, -0.021982777, 0.287149291, -1.405635476, -0.002911264, -0.394575970)
names_coeffs <- c("intercept", "M1", "M2", "MlogR", "logR", "R", "logVS")
phi_s2s_sim <- 0.43
tau_sim_val <- c(0.4,0.25)
phi_sim_val <- c(0.55,0.4)
mb_tau <- c(5,6)
mb_phi <- c(4.5,5.5)p1 <- data.frame(res_phi, res_phi_stan[,c(1,2)], res_phi_stan2[,c(1,2)]) %>%
set_names(c('sd(dWS)_lowm','sd(dWS)+unc_lowm','sd(dWS)_largem','sd(dWS)+unc_largem',
'stan_lowm','stan_largem','stanf_lowm','stanf_largem')) %>%
pivot_longer(everything(), names_to = c('model','mag'),names_sep = '_') %>%
ggplot() +
geom_density(aes(x = value, color = model, linetype = mag), linewidth = lw, key_glyph = draw_key_path) +
geom_vline(xintercept = phi_sim_val[2], linewidth = lw) +
geom_vline(xintercept = phi_sim_val[1], linetype = 'dashed', linewidth = lw) +
scale_color_manual(values = c('orange','red','blue','cyan'),
labels = c(TeX("sd(\\widehat{\\delta WS})"),
TeX("sd(\\widehat{\\delta WS} + unc)"),
TeX('stan ($\\delta R$)'), 'stan (full)')) +
scale_linetype_manual(values = c(1,2),
labels = c(TeX(sprintf("$M \\geq %.1f$",mb_phi[2])),
TeX(sprintf("$M \\leq %.1f$",mb_phi[1])))) +
guides(color = guide_legend(title = NULL), linetype = guide_legend(title = NULL)) +
theme(legend.position = c(0.5,0.8),
legend.key.width = unit(2,'cm')) +
labs(x = expression(paste(widehat(phi)[SS]))) +
lims(y = c(0,100))
p2 <- data.frame(res_tau, res_tau_stan[,c(1,2)], res_tau_stan2[,c(1,2)]) %>%
set_names(c('sd(dB)_lowm','sd(dB)+unc_lowm','sd(dB)_largem','sd(dB)+unc_largem',
'stan_lowm','stan_largem','stanf_lowm','stanf_largem')) %>%
pivot_longer(everything(), names_to = c('model','mag'),names_sep = '_') %>%
ggplot() +
geom_density(aes(x = value, color = model, linetype = mag), linewidth = lw, key_glyph = draw_key_path) +
geom_vline(xintercept = tau_sim_val[2], linewidth = lw) +
geom_vline(xintercept = tau_sim_val[1], linetype = 'dashed', linewidth = lw) +
scale_color_manual(values = c('orange','red','blue','cyan'),
labels = c(TeX("sd(\\widehat{\\delta B})"),
TeX("sd(\\widehat{\\delta B} + unc)"),
TeX('stan ($\\delta R$)'), 'stan (full)')) +
scale_linetype_manual(values = c(1,2),
labels = c(TeX(sprintf("$M \\geq %.1f$",mb_tau[2])),
TeX(sprintf("$M \\leq %.1f$",mb_tau[1])))) +
guides(color = guide_legend(title = NULL), linetype = guide_legend(title = NULL)) +
theme(legend.position = 'none') +
labs(x = expression(paste(widehat(tau))))
p3 <- data.frame(res_phis2s[,c(1,3)], res_phis2s_stan2[,1]) %>%
set_names(c('lmer','stan','stanf')) %>%
pivot_longer(everything()) %>%
ggplot() +
geom_density(aes(x = value, color = name), linewidth = lw, key_glyph = draw_key_path) +
geom_vline(xintercept = phi_s2s_sim, linewidth = lw) +
scale_color_manual(values = c('red','blue','cyan'),
labels = c('lmer', TeX('stan ($\\delta R$)'), 'stan (full')) +
guides(color = guide_legend(title = NULL)) +
theme(legend.position = 'none') +
labs(x = expression(paste(widehat(phi)[S2S])))
leg <- ggpubr::get_legend(p1)
patchwork::wrap_plots(p1 + theme(legend.position = 'none'),p2,p3,ggpubr::as_ggplot(leg), ncol = 2)
df1 <- data.frame(res_coeffs) %>% set_names(names_coeffs)
df1$model <- 'lmer'
df2 <- data.frame(res_coeffs_stan2) %>% set_names(names_coeffs)
df2$model <- 'stan'
df <- data.frame(name = names_coeffs,
true = coeffs)
rbind(df1 %>% pivot_longer(!model),
df2 %>% pivot_longer(!model)) %>%
ggplot() +
geom_density(aes(x = value, color = model), linewidth = 1.5, key_glyph = draw_key_path) +
facet_wrap(vars(name), scales = "free") +
geom_vline(aes(xintercept = true), data = df, linewidth = 1.5) +
guides(color = guide_legend(title = NULL)) +
labs(x = '') +
theme(legend.position = c(0.8,0.2),
strip.text = element_text(size = 20))
4.3 \(V_{S30}\)-Scaling from Site Terms
Here, we show results of estimating the coefficient for the \(V_{S30}\)-scaling from the full model, site terms, and a mixed-effects regression of total residuals. We performed the repeated simulations for both the CB141 data and the Ialian data, and below we show results for both. We see very similar results, with a larger bias for the estimaton from well-recorded stations (more than 9 records) for the Italian data. There are less such stations in the Italian data set, which results in this larger bias. This is a reminder that the size of the biases that can occur depends on the data set.
set1 <- RColorBrewer::brewer.pal(7, "Set1")
coeff_vs <- -0.394575970
# Results for simulations based on CB14 data
load(file = file.path('./Git/MixedModels_Biases/', 'results', 'res_vs_ita18_CB.Rdata'))
xlab <- expression(paste(c[vs]))
names <- c('full','dS','dS(N>10)','dR')
df <- data.frame(res_val[,c(1,5,10,15)]) %>% set_names(names) %>%
pivot_longer(everything())
df$name <- factor(df$name, names)
p1 <- ggplot(df) +
geom_density(aes(x = value, color = name), linewidth = 1.5, key_glyph = draw_key_path) +
geom_vline(xintercept = coeff_vs, linewidth = 1.5) +
scale_color_manual(values=set1,
labels = c('full',TeX('$\\widehat{\\delta S}$'),
TeX('$\\widehat{\\delta S}$ ($N \\geq 10$)'),
TeX('$\\widehat{\\delta R}$'))) +
guides(color = guide_legend(title=NULL)) +
labs(x = xlab, title = 'CB14 Data') +
theme(legend.position = c(0.2,0.8))
# Results for simulations based on Italian
load(file = file.path('./Git/MixedModels_Biases/', 'results', 'res_vs_ita18_italy.Rdata'))
xlab <- expression(paste(c[vs]))
names <- c('full','dS','dS(N>10)','dR')
df <- data.frame(res_val[,c(1,5,10,15)]) %>% set_names(names) %>%
pivot_longer(everything())
df$name <- factor(df$name, names)
p2 <- ggplot(df) +
geom_density(aes(x = value, color = name), linewidth = 1.5, key_glyph = draw_key_path) +
geom_vline(xintercept = coeff_vs, linewidth = 1.5) +
scale_color_manual(values=set1,
labels = c('full',TeX('$\\widehat{\\delta S}$'),
TeX('$\\widehat{\\delta S}$ ($N \\geq 10$)'),
TeX('$\\widehat{\\delta R}$'))) +
guides(color = guide_legend(title=NULL)) +
labs(x = xlab, title = 'Italian Data') +
theme(legend.position = c(0.2,0.8))
patchwork::wrap_plots(p1, p2)
4.4 Correlations of Random Effects
tau_sim1 <- 0.4
phi_s2s_sim1 <- 0.43
phi_ss_sim1 <- 0.5
tau_sim2 <- 0.45
phi_s2s_sim2 <- 0.4
phi_ss_sim2 <- 0.55
sigma_tot1 <- sqrt(tau_sim1^2 + phi_s2s_sim1^2 + phi_ss_sim1^2)
sigma_tot2 <- sqrt(tau_sim2^2 + phi_s2s_sim2^2 + phi_ss_sim2^2)load(file.path('./Git/MixedModels_Biases/', 'results',
'res_corrre_CB14_high.Rdata'))
rho_tau <- 0.95
rho_ss <- 0.9
rho_s2s <- 0.85
rho_total <- (rho_tau * tau_sim1 * tau_sim2 +
rho_s2s * phi_s2s_sim1 * phi_s2s_sim2 +
rho_ss * phi_ss_sim1 * phi_ss_sim2) /
(sigma_tot1 * sigma_tot2)
rho_total_sample <- (mat_cor_sample[,2] * tau_sim1 * tau_sim2 +
mat_cor_sample[,1] * phi_s2s_sim1 * phi_s2s_sim2 +
mat_cor_sample[,3] * phi_ss_sim1 * phi_ss_sim2) /
(sigma_tot1 * sigma_tot2)
patchwork::wrap_plots(
data.frame(mat_cor[,c(1,6)]) %>%
pivot_longer(everything()) %>%
ggplot() +
geom_density(data.frame(x = mat_cor_sample[,1]),
mapping = aes(x = x), color = 'gray', linewidth = lw) +
geom_density(aes(x = value, color = name), linewidth = lw, key_glyph = draw_key_path) +
geom_vline(xintercept = rho_s2s, linewidth = lw) +
labs(x = TeX('$\\rho(\\delta S_1, \\delta S_2)$')) +
scale_color_manual(values = c('red','blue'),
labels = c(TeX('$\\rho_{sample}$'),
TeX('$cov_{sample}/(\\hat{\\sigma}_{1} \\; \\hat{\\sigma}_{2})$'))) +
guides(color = guide_legend(title = NULL)) +
theme(legend.position = c(0.4,0.8),
legend.key.width = unit(2,'cm')) +
#lims(x = c(0.25,0.6)),
lims(x = c(0.45,0.95)),
data.frame(mat_cor[,c(2,7)]) %>%
pivot_longer(everything()) %>%
ggplot() +
geom_density(data.frame(x = mat_cor_sample[,2]),
mapping = aes(x = x), color = 'gray', linewidth = lw) +
geom_density(aes(x = value, color = name), linewidth = lw, key_glyph = draw_key_path) +
geom_vline(xintercept = rho_tau, linewidth = lw) +
labs(x = TeX('$\\rho(\\delta B_1, \\delta B_2)$')) +
scale_color_manual(values = c('red','blue'),
labels = c(TeX('$\\rho_{sample}$'),'re2')) +
guides(color = guide_legend(title = NULL)) +
theme(legend.position = 'none',
legend.key.width = unit(2,'cm')) +
#lims(x = c(0.3,0.6)),
lims(x = c(0.8,1.0)),
data.frame(mat_cor[,c(3,8)]) %>%
pivot_longer(everything()) %>%
ggplot() +
geom_density(data.frame(x = mat_cor_sample[,3]),
mapping = aes(x = x), color = 'gray', linewidth = lw) +
geom_density(aes(x = value, color = name), linewidth = lw, key_glyph = draw_key_path) +
geom_vline(xintercept = rho_ss, linewidth = lw) +
labs(x = TeX('$\\rho(\\delta WS_1, \\delta WS_2)$')) +
scale_color_manual(values = c('red','blue'),
labels = c('re','re2')) +
guides(color = guide_legend(title = NULL)) +
theme(legend.position = 'none',
legend.key.width = unit(2,'cm')) +
#lims(x = c(0.4,0.55)),
lims(x = c(0.8,0.95)),
data.frame(mat_cor[,c(4,5,9)]) %>%
pivot_longer(everything()) %>%
ggplot() +
geom_density(data.frame(x = rho_total_sample),
mapping = aes(x = x), color = 'gray', linewidth = lw) +
geom_density(aes(x = value, color = name), linewidth = lw, key_glyph = draw_key_path) +
geom_vline(xintercept = rho_total, linewidth = lw) +
labs(x = TeX('$\\rho_{total}$')) +
scale_color_manual(values = c('red','blue','cyan'),
labels = c("(1)","(2)",'(3)')) +
guides(color = guide_legend(title = NULL)) +
theme(legend.position = c(0.2,0.8),
legend.key.width = unit(2,'cm')) +
#lims(x = c(0.45,0.55))
lims(x = c(0.875,0.925))
)
func_ci <- function(cor, n, rho) {
r_fisher <- log((1+cor) / (1-cor)) / 2
r_lb <- r_fisher - (1.64 /sqrt(n - 3))
r_ub <- r_fisher + (1.64 /sqrt(n - 3))
cor_lme_lb <- (exp(2 * r_lb) - 1) / (exp(2 * r_lb) + 1)
cor_lme_ub <- (exp(2 * r_ub) - 1) / (exp(2 * r_ub) + 1)
return(sum(cor_lme_lb <= rho & cor_lme_ub >= rho) / length(cor))
}
n_eq <- 274
n_stat <- 1519
n_rec <- 12482
knitr::kable(data.frame(dS = c(func_ci(mat_cor_sample[,1], n_stat, rho_s2s), func_ci(mat_cor[,1], n_stat, rho_s2s)),
dB = c(func_ci(mat_cor_sample[,2], n_eq, rho_tau), func_ci(mat_cor[,2], n_eq, rho_tau)),
dWS = c(func_ci(mat_cor_sample[,3], n_rec, rho_ss), func_ci(mat_cor[,3], n_rec, rho_ss)),
row.names = c('simulated','estimated')),
row.names = TRUE,
caption = "Fraction of correlation coefficiets inside 90% confidence interval."
)| dS | dB | dWS | |
|---|---|---|---|
| simulated | 0.925 | 0.895 | 0.890 |
| estimated | 0.360 | 0.860 | 0.785 |
load(file.path('./Git/MixedModels_Biases/', 'results',
'res_corrre_CB14_low.Rdata'))
rho_tau <- 0.45
rho_ss <- 0.5
rho_s2s <- 0.55
rho_total <- (rho_tau * tau_sim1 * tau_sim2 +
rho_s2s * phi_s2s_sim1 * phi_s2s_sim2 +
rho_ss * phi_ss_sim1 * phi_ss_sim2) /
(sigma_tot1 * sigma_tot2)
rho_total_sample <- (mat_cor_sample[,2] * tau_sim1 * tau_sim2 +
mat_cor_sample[,1] * phi_s2s_sim1 * phi_s2s_sim2 +
mat_cor_sample[,3] * phi_ss_sim1 * phi_ss_sim2) /
(sigma_tot1 * sigma_tot2)
patchwork::wrap_plots(
data.frame(mat_cor[,c(1,6)]) %>%
pivot_longer(everything()) %>%
ggplot() +
geom_density(data.frame(x = mat_cor_sample[,1]),
mapping = aes(x = x), color = 'gray', linewidth = lw) +
geom_density(aes(x = value, color = name), linewidth = lw, key_glyph = draw_key_path) +
geom_vline(xintercept = rho_s2s, linewidth = lw) +
labs(x = TeX('$\\rho(\\delta S_1, \\delta S_2)$')) +
scale_color_manual(values = c('red','blue'),
labels = c(TeX('$\\rho_{sample}$'),
TeX('$cov_{sample}/(\\hat{\\sigma}_{1} \\; \\hat{\\sigma}_{2})$'))) +
guides(color = guide_legend(title = NULL)) +
theme(legend.position = c(0.4,0.8),
legend.key.width = unit(2,'cm')) +
lims(x = c(0.25,0.6)),
data.frame(mat_cor[,c(2,7)]) %>%
pivot_longer(everything()) %>%
ggplot() +
geom_density(data.frame(x = mat_cor_sample[,2]),
mapping = aes(x = x), color = 'gray', linewidth = lw) +
geom_density(aes(x = value, color = name), linewidth = lw, key_glyph = draw_key_path) +
geom_vline(xintercept = rho_tau, linewidth = lw) +
labs(x = TeX('$\\rho(\\delta B_1, \\delta B_2)$')) +
scale_color_manual(values = c('red','blue'),
labels = c(TeX('$\\rho_{sample}$'),'re2')) +
guides(color = guide_legend(title = NULL)) +
theme(legend.position = 'none',
legend.key.width = unit(2,'cm')) +
lims(x = c(0.3,0.6)),
data.frame(mat_cor[,c(3,8)]) %>%
pivot_longer(everything()) %>%
ggplot() +
geom_density(data.frame(x = mat_cor_sample[,3]),
mapping = aes(x = x), color = 'gray', linewidth = lw) +
geom_density(aes(x = value, color = name), linewidth = lw, key_glyph = draw_key_path) +
geom_vline(xintercept = rho_ss, linewidth = lw) +
labs(x = TeX('$\\rho(\\delta WS_1, \\delta WS_2)$')) +
scale_color_manual(values = c('red','blue'),
labels = c('re','re2')) +
guides(color = guide_legend(title = NULL)) +
theme(legend.position = 'none',
legend.key.width = unit(2,'cm')) +
lims(x = c(0.4,0.55)),
data.frame(mat_cor[,c(4,5,9)]) %>%
pivot_longer(everything()) %>%
ggplot() +
geom_density(aes(x = value, color = name), linewidth = lw, key_glyph = draw_key_path) +
geom_vline(xintercept = rho_total, linewidth = lw) +
labs(x = TeX('$\\rho_{total}$')) +
scale_color_manual(values = c('red','blue','cyan'),
labels = c("(1)","(2)",'(3)')) +
guides(color = guide_legend(title = NULL)) +
theme(legend.position = c(0.2,0.8),
legend.key.width = unit(2,'cm')) +
lims(x = c(0.45,0.55))
)## Warning: Removed 2 rows containing non-finite outside the scale range
## (`stat_density()`).## Warning: Removed 3 rows containing non-finite outside the scale range
## (`stat_density()`).
knitr::kable(data.frame(dS = c(func_ci(mat_cor_sample[,1], n_stat, rho_s2s), func_ci(mat_cor[,1], n_stat, rho_s2s)),
dB = c(func_ci(mat_cor_sample[,2], n_eq, rho_tau), func_ci(mat_cor[,2], n_eq, rho_tau)),
dWS = c(func_ci(mat_cor_sample[,3], n_rec, rho_ss), func_ci(mat_cor[,3], n_rec, rho_ss)),
row.names = c('simulated','estimated')),
row.names = TRUE,
caption = "Fraction of correlation coefficiets inside 90% confidence interval."
)| dS | dB | dWS | |
|---|---|---|---|
| simulated | 0.925 | 0.895 | 0.88 |
| estimated | 0.775 | 0.895 | 0.88 |
4.5 Correlations with Stress Drop
# Results for simulations based on CB14 data
df_res_cor <- read.csv(file.path('./Git/MixedModels_Biases/', 'results',
'res_sim_cor_CB_N50.csv'))
tau_sim <- 0.4
phi_s2s_sim <- 0.43
phi_ss_sim <- 0.5
rho <- 0.7
tau2 <- 1
df_res_cor %>% pivot_longer(c(cor_sim, cor_lme, cor_mean)) %>%
ggplot() +
geom_density(aes(x = value, color = name), linewidth = lw, key_glyph = draw_key_path) +
geom_vline(aes(xintercept = rho), linewidth = lw) +
scale_color_manual(values=c("blue", 'red', 'gray'),
labels=c("2-step lmer", "1-step stan", "sim")
) +
guides(color = guide_legend(title=NULL)) +
theme(legend.position = c(0.18,0.89)) +
labs(x = expression(widehat(rho)))
knitr::kable(data.frame(simulated = func_ci(df_res_cor$cor_sim, n_eq, rho),
Stan = (sum(df_res_cor$cor_q05 <= rho & df_res_cor$cor_q95 >= rho)) / nrow(df_res_cor),
lmer = func_ci(df_res_cor$cor_lme, n_eq, rho)),
row.names = FALSE,
caption = "Fraction of correlation coefficiets inside 90% confidence interval."
)| simulated | Stan | lmer |
|---|---|---|
| 0.94 | 0.98 | 0.52 |
4.6 Spatial Correlations of Site Terms
seed <- 1701
load(file = file.path('./Git/MixedModels_Biases/', 'results', sprintf('res_spatial_ita18_italy_seed%d.Rdata', seed)))
load(file = file.path('./Git/MixedModels_Biases/', 'results', sprintf('res_spatial_ita18b_italy_seed%d.Rdata', seed)))
range <- 30
wvar <- 0.65
tau_sim <- 0.17
phi_s2s_sim <- 0.23
phi_s2s_0 <- sqrt((1 - wvar) * phi_s2s_sim^2)
phi_s2s_c <- sqrt(wvar * phi_s2s_sim^2)
phi_ss_sim <- 0.2
p1 <- data.frame(res_spatial[,c(4,7)],res_spatial_tot[,4]) %>% set_names('m1','m3','m2') %>%
pivot_longer(everything()) %>%
ggplot() +
geom_density(aes(x = value, color = name), linewidth = lw, key_glyph = draw_key_path) +
geom_vline(aes(xintercept = range), linewidth = lw) +
labs(x = 'spatial range (km)') +
theme(legend.position = c(0.85,0.85)) +
guides(color = guide_legend(title=NULL)) +
scale_color_manual(values = c('blue','red','orange'),
labels = c('full',
TeX("$\\widehat{\\delta R}$"),
TeX("$\\widehat{\\delta S}$")))
p2 <- data.frame(m1 = res_spatial[,5]^2 / (1/res_spatial[,3] + res_spatial[,5]^2),
m1 = res_spatial_tot[,5]^2 / (1/res_spatial_tot[,3] + res_spatial_tot[,5]^2),
m3 = res_spatial[,8]^2 / (1/res_spatial[,6] + res_spatial[,8]^2)) %>%
pivot_longer(everything()) %>%
ggplot() +
geom_density(aes(x = value, color = name), linewidth = 1.5, key_glyph = draw_key_path) +
geom_vline(aes(xintercept = wvar), linewidth = 1.5) +
labs(x = TeX("$\\hat{\\phi}_{S2S,c}^2 / \\hat{phi}_{S2S}^2$")) +
theme(legend.position = 'none') +
guides(color = guide_legend(title=NULL)) +
scale_color_manual(values = c('blue','red','orange'),
labels = c('full',
TeX("$\\widehat{\\delta R}$"),
TeX("$\\widehat{\\delta S}$")))
p3 <- data.frame(res_spatial[,c(5,8)],res_spatial_tot[,5]) %>% set_names('m1','m3','m2') %>%
pivot_longer(everything()) %>%
ggplot() +
geom_density(aes(x = value, color = name), linewidth = lw, key_glyph = draw_key_path) +
geom_vline(aes(xintercept = phi_s2s_c), linewidth = lw) +
labs(x = TeX("$\\hat{\\phi}_{S2S,c}$")) +
theme(legend.position = 'none') +
guides(color = guide_legend(title=NULL)) +
scale_color_manual(values = c('blue','red','orange'),
labels = c('full',
TeX("$\\widehat{\\delta R}$"),
TeX("$\\widehat{\\delta S}$")))
p4 <- data.frame(1/sqrt(res_spatial[,c(3,6)]),1/sqrt(res_spatial_tot[,3])) %>% set_names('m1','m3','m2') %>%
pivot_longer(everything()) %>%
ggplot() +
geom_density(aes(x = value, color = name), linewidth = lw, key_glyph = draw_key_path) +
geom_vline(aes(xintercept = phi_s2s_0), linewidth = lw) +
labs(x = TeX("$\\hat{\\phi}_{S2S,0}$")) +
theme(legend.position = 'none') +
guides(color = guide_legend(title=NULL)) +
scale_color_manual(values = c('blue','red','orange'),
labels = c('full',
TeX("$\\widehat{\\delta R}$"),
TeX("$\\widehat{\\delta S}$")))
patchwork::wrap_plots(p1,p2,p3,p4, ncol = 2)
4.7 Cell-Specific Attenuation
tau <- 0.17
phi_s2s <- 0.2
phi_0 <- 0.18
sigma_cell <- 0.35
coeffs <- c(3.421046409, 0.193954090, -0.021982777, 0.287149291, -1.405635476,
-0.002911264, 0.085983743, 0.010500239, -0.394575970)
seed <- 5618
load(file = file.path('./Git/MixedModels_Biases/', 'results',
sprintf('res_cell_italy_ita18_seed%d.Rdata', seed)))
p1 <- data.frame(res_cell[,c(4,10,6)]) %>% set_names(c('full','dR','dWS')) %>%
pivot_longer(everything()) %>%
ggplot() +
geom_density(aes(x = value, color = name), linewidth = 1.5, key_glyph = draw_key_path) +
geom_vline(aes(xintercept = sigma_cell), linewidth = 1.5) +
labs(x = expression(paste(sigma[cell]))) +
theme(legend.position = c(0.5,0.85)) +
guides(color = guide_legend(title=NULL)) +
scale_color_manual(values = c('red','orange','blue'),
labels = c(TeX("$\\delta R$"), TeX("$\\widehat{\\delta WS}$"), "full"))
p2 <- data.frame(res_cell_fix[,c(1,7,4)]) %>% set_names(c('full','dR','dWS')) %>%
pivot_longer(everything()) %>%
ggplot() +
geom_density(aes(x = value, color = name), linewidth = lw, key_glyph = draw_key_path) +
geom_vline(aes(xintercept = coeffs[6]), linewidth = lw) +
labs(x = expression(paste(c[attn]))) +
theme(legend.position = 'none') +
guides(color = guide_legend(title=NULL)) +
scale_color_manual(values = c('red','orange','blue'),
labels = c(TeX("$\\delta R$"), TeX("$\\widehat{\\delta WS}$"), "full"))
p3 <- data.frame(full = res_cell[,1]^2/res_cell_sdlme[,3]^2,
dR = res_cell[,7]^2/res_cell_sdlme[,3]^2,
dS = res_cell[,5]^2/res_cell_sdlme[,3]^2) %>%
pivot_longer(everything()) %>%
ggplot() +
geom_density(aes(x = value, color = name), linewidth = 1.5) +
labs(x = TeX("$\\hat{\\phi}_{SS,0}^2 / \\hat{\\phi}_{SS}^2$")) +
theme(legend.position = 'none') +
guides(color = guide_legend(title=NULL)) +
scale_color_manual(values = c('red','orange','blue'),
labels = c(TeX("$\\delta R$"), TeX("$\\widehat{\\delta WS}$"), "full"))
p4 <- data.frame(res_cell_cor[,c(4,6,5)]) %>%
set_names(c('full','dR','dWS')) %>%
pivot_longer(everything()) %>%
ggplot() +
geom_density(aes(x = value, color = name), linewidth = 1.5) +
labs(x = expression(paste(rho,'(c'[true],',c'[est],')'))) +
theme(legend.position = 'none') +
guides(color = guide_legend(title=NULL)) +
scale_color_manual(values = c('red','orange','blue'),
labels = c(TeX("$\\delta R$"), TeX("$\\widehat{\\delta WS}$"), "full"))
patchwork::wrap_plots(p1,p2,p3,p4, ncol = 2)